The following sequence of polynomials with positive integer coefficients arises in a physics problem related to the quantum harmonic oscillator:
\begin{align} p_{0}(x) &= 1 \\ p_{1}(x) &= x+1 \\ p_{2}(x) &= x^2+2 x+2 \\ p_{3}(x) &= x^3+3 x^2+8 x+6 \\ p_{4}(x) &= x^4+4 x^3+20 x^2+32 x+24 \\ p_{5}(x) &= x^5+5 x^4+40 x^3+100 x^2+184 x+120 \\ p_{6}(x) &= x^6+6 x^5+70 x^4+240 x^3+784 x^2+1104 x+720 \\ p_{7}(x) &= x^7+7 x^6+112 x^5+490 x^4+2464 x^3+5488 x^2+8448 x+5040 \\ p_{8}(x) &= x^8+8 x^7+168 x^6+896 x^5+6384 x^4+19712 x^3+52352 \ x^2+67584 x+40320 \end{align}
There are some interesting patterns in their coefficients, but despite my best efforts (and numerous OEIS lookups) I haven't been able to pin down a closed form or recurrence relation. Does anybody see one of these (or a similar nice characterization of $p_n$)?
The coefficients of $p_0, \dots, p_{30}$, which are too big to be nicely $\mathrm{\LaTeX}$'ed, are provided in the following table:
+---+----+------+--------+----------+-----------+-------------+---------------+----------------+------------------+-------------------+---------------------+----------------------+-----------------------+-------------------------+--------------------------+---------------------------+----------------------------+-----------------------------+------------------------------+-------------------------------+--------------------------------+---------------------------------+---------------------------------+----------------------------------+----------------------------------+-----------------------------------+-----------------------------------+-----------------------------------+-----------------------------------+-----------------------------------+
| 1 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
| 1 | 1 | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
| 1 | 2 | 2 | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
| 1 | 3 | 8 | 6 | | | | | | | | | | | | | | | | | | | | | | | | | | | |
| 1 | 4 | 20 | 32 | 24 | | | | | | | | | | | | | | | | | | | | | | | | | | |
| 1 | 5 | 40 | 100 | 184 | 120 | | | | | | | | | | | | | | | | | | | | | | | | | |
| 1 | 6 | 70 | 240 | 784 | 1104 | 720 | | | | | | | | | | | | | | | | | | | | | | | | |
| 1 | 7 | 112 | 490 | 2464 | 5488 | 8448 | 5040 | | | | | | | | | | | | | | | | | | | | | | | |
| 1 | 8 | 168 | 896 | 6384 | 19712 | 52352 | 67584 | 40320 | | | | | | | | | | | | | | | | | | | | | | |
| 1 | 9 | 240 | 1512 | 14448 | 57456 | 229760 | 471168 | 648576 | 362880 | | | | | | | | | | | | | | | | | | | | | |
| 1 | 10 | 330 | 2400 | 29568 | 144480 | 804320 | 2297600 | 5360256 | 6485760 | 3628800 | | | | | | | | | | | | | | | | | | | | |
| 1 | 11 | 440 | 3630 | 55968 | 325248 | 2393600 | 8847520 | 30633856 | 58962816 | 74972160 | 39916800 | | | | | | | | | | | | | | | | | | | |
| 1 | 12 | 572 | 5280 | 99528 | 671616 | 6296576 | 28723200 | 136804096 | 367606272 | 782525952 | 899665920 | 479001600 | | | | | | | | | | | | | | | | | | |
| 1 | 13 | 728 | 7436 | 168168 | 1293864 | 15027584 | 81855488 | 510205696 | 1778453248 | 5561407488 | 10172837376 | 12174658560 | 6227020800 | | | | | | | | | | | | | | | | | |
| 1 | 14 | 910 | 10192 | 272272 | 2354352 | 33141680 | 210386176 | 1656182528 | 7142879744 | 30459752960 | 77859704832 | 154594381824 | 170445219840 | 87178291200 | | | | | | | | | | | | | | | | |
| 1 | 15 | 1120 | 13650 | 425152 | 4084080 | 68456960 | 497125200 | 4811975168 | 24842737920 | 137602949120 | 456896294400 | 1322489954304 | 2318915727360 | 2643856588800 | 1307674368000 | | | | | | | | | | | | | | | |
| 1 | 16 | 1360 | 17920 | 643552 | 6802432 | 133802240 | 1095311360 | 12765978368 | 76991602688 | 535086755840 | 2201647185920 | 8632830664704 | 21159839268864 | 39746508226560 | 42301705420800 | 20922789888000 | | | | | | | | | | | | | | |
| 1 | 17 | 1632 | 23120 | 948192 | 10940384 | 249443584 | 2274638080 | 31386271488 | 217021632256 | 1843944001536 | 9096474849280 | 46060832825344 | 146758121299968 | 399463775797248 | 675690639851520 | 740051782041600 | 355687428096000 | | | | | | | | | | | | | |
| 1 | 18 | 1938 | 29376 | 1364352 | 17067456 | 446370496 | 4489984512 | 72329756928 | 564952886784 | 5750333382144 | 33190992027648 | 209797380112384 | 829094990856192 | 3041109959196672 | 7190347964350464 | 12902483299368960 | 13320932076748800 | 6402373705728000 | | | | | | | | | | | | |
| 1 | 19 | 2280 | 36822 | 1922496 | 25922688 | 770652160 | 8481039424 | 157639462656 | 1374265381632 | 16484438231040 | 109256334260736 | 840426228637696 | 3986150222135296 | 18793914785464320 | 57781089224736768 | 149519094622027776 | 245147182688010240 | 259500083163955200 | 121645100408832000 | | | | | | | | | | | |
| 1 | 20 | 2660 | 45600 | 2658936 | 38449920 | 1289105920 | 15413043200 | 327260251136 | 3152789253120 | 43969745863680 | 329688764620800 | 3025552913852416 | 16808524572753920 | 98516919228170240 | 375878295709286400 | 1305140879116763136 | 2990381892440555520 | 5162443736924160000 | 5190001663279104000 | 2432902008176640000 | | | | | | | | | | |
| 1 | 21 | 3080 | 55860 | 3616536 | 55837656 | 2096554240 | 27071224320 | 650934158336 | 6872465273856 | 110178320179200 | 923364663137280 | 9949016970889216 | 63536611190900736 | 451495935256002560 | 2068855303791575040 | 9198585089011777536 | 27407958461452025856 | 67960463478175825920 | 108411318475407360000 | 111422936937037824000 | 51090942171709440000 | | | | | | | | | |
| 1 | 22 | 3542 | 67760 | 4845456 | 79563792 | 3324982672 | 46124193280 | 1246501093376 | 14320551483392 | 261372556204032 | 2423923043942400 | 30263039559909376 | 218878373359562752 | 1849301381455818752 | 9932910575632056320 | 54713401772426428416 | 202368871958259105792 | 670935549630120394752 | 1495130196519868170240 | 2496471943395999744000 | 2451304612614832128000 | 1124000727777607680000 | | | | | | | | |
| 1 | 23 | 4048 | 81466 | 6403936 | 111445488 | 5154949888 | 76474601456 | 2307357538816 | 28669525147648 | 590745240322048 | 6011568792692736 | 86013269570584576 | 696049909877915648 | 6883503968725762048 | 42533931773483831296 | 283170345011963723776 | 1258408240765807853568 | 5325419604670079827968 | 15431517641492769079296 | 36884466463352967659520 | 57418854698107994112000 | 57504006817918746624000 | 25852016738884976640000 | | | | | | | |
| 1 | 24 | 4600 | 97152 | 8359120 | 153694464 | 7829641600 | 123718797312 | 4142747973760 | 55376580931584 | 1278813843865600 | 14177885767729152 | 230290920595210240 | 2064318469694029824 | 23588701805795737600 | 165204095249418289152 | 1303984707575575674880 | 6796088280287129370624 | 35527217383049468313600 | 127810070512081915871232 | 407240889859179425562624 | 885227195120471223828480 | 1435556519572510605312000 | 1380096163630049918976000 | 620448401733239439360000 | | | | | | |
| 1 | 25 | 5200 | 115000 | 10787920 | 208978000 | 11672003200 | 195741040000 | 7235717906560 | 103568699344000 | 2663228367155200 | 31970346096640000 | 584738064788439040 | 5757273014880256000 | 75196663548146483200 | 589717545144893440000 | 5434087088811032903680 | 32599617689389391872000 | 205429424390227702579200 | 888180434576236707840000 | 3602492652661227322343424 | 10181022246479485639065600 | 23566236397584291201024000 | 35888912989312765132800000 | 35122852492484487413760000 | 15511210043330985984000000 | | | | | |
| 1 | 26 | 5850 | 135200 | 13777920 | 280485920 | 17105431200 | 303472083200 | 12324984946560 | 188128665570560 | 5356014550099200 | 69243937546035200 | 1415967063301079040 | 15203189684499415040 | 224885761935033139200 | 1955113252251808563200 | 20766743262578262343680 | 141286264309086855495680 | 1053019484314351891251200 | 5341165034145920267059200 | 26695183951643381726183424 | 93664808969191910380929024 | 288272814806050917816729600 | 612722146337191571226624000 | 968234590214616380866560000 | 913194164804596672757760000 | 403291461126605635584000000 | | | | |
| 1 | 27 | 6552 | 157950 | 17428320 | 372003840 | 24678551040 | 461846642400 | 20518731192960 | 332774593557120 | 10435488520504320 | 144612392852678400 | 3285553377044029440 | 38231110709129134080 | 635371883416517345280 | 6071915572245894758400 | 73554801073377093550080 | 560702068089613083279360 | 4867748620659696989634560 | 28431526076487501063782400 | 170906639873583228936781824 | 720769966694371306606952448 | 2817222656974232498101813248 | 7783365999763374781051699200 | 17511732541318788803985408000 | 26142333935794642283397120000 | 25059533910850715800043520000 | 10888869450418352160768000000 | | | |
| 1 | 28 | 7308 | 183456 | 21850920 | 487992960 | 35094658560 | 690999429120 | 33450437180160 | 574524473402880 | 19753177140103680 | 292193678574120960 | 7334700376919024640 | 91995494557232824320 | 1705842983272133099520 | 17790412735662485667840 | 243568437096262146785280 | 2059534430054558619402240 | 20567406527168863321456640 | 136296961378471515709767680 | 966989370015233258722689024 | 4785385916460330410229891072 | 22998781724416629083096481792 | 78882234395278509946850770944 | 235445980534693282673432985600 | 490328511156926086511591424000 | 757044884113100699735162880000 | 701666949503820042401218560000 | 304888344611713860501504000000 | | |
| 1 | 29 | 8120 | 211932 | 27171144 | 633676680 | 49246454400 | 1017745098240 | 53489420624640 | 970062678224640 | 36414386868787200 | 572842137063006720 | 15808317055568532480 | 212706310930651714560 | 4373712325431885004800 | 49469446514891859886080 | 759490406430474231152640 | 7063484675791602256773120 | 80293904998751063284121600 | 596454789287897036322242560 | 4919601249990907214305951744 | 28042691730441764502957981696 | 161774973301766210979763322880 | 666964670008082243409797971968 | 2523030777997770071132105342976 | 6827933435506105197529556582400 | 14976571707663957208571314176000 | 21954301639279920292319723520000 | 20653229880222495090136842240000 | 8841761993739701954543616000000 | |
| 1 | 30 | 8990 | 243600 | 33529104 | 815134320 | 68256754800 | 1477393632000 | 84021773571840 | 1604682618739200 | 65516267215526400 | 1092431606063616000 | 32993581167458734080 | 474249511667055974400 | 10754901653351436441600 | 131211369762956550144000 | 2243573801877230027735040 | 22784712192914226934579200 | 292198445272499130987315200 | 2408817149962531898523648000 | 22813244928627818303973228544 | 147588037499727216429178552320 | 1003055725215019146068502773760 | 4853249199052986329392899686400 | 22531970878240237373426044502016 | 75690923339933102133963160289280 | 219814574772847113134458011648000 | 449297151229918716257139425280000 | 679282279058620103859728547840000 | 619596896406674852704105267200000 | 265252859812191058636308480000000 |
+---+----+------+--------+----------+-----------+-------------+---------------+----------------+------------------+-------------------+---------------------+----------------------+-----------------------+-------------------------+--------------------------+---------------------------+----------------------------+-----------------------------+------------------------------+-------------------------------+--------------------------------+---------------------------------+---------------------------------+----------------------------------+----------------------------------+-----------------------------------+-----------------------------------+-----------------------------------+-----------------------------------+-----------------------------------+
With the help of Jyrki Lahtonen's comments, it turns out that these polynomials are a shifted version of the Meixner polynomials $M_n(x)$, defined by the recurrence relation
$$ M_{-1}(x) = 0 \qquad M_0(x) = 1 \qquad M_{n+1}(x) =xM_n(x)-n^2M_{n-1}(x) $$
In particular,
$$ p_n(x) = \frac{M_n(ix+i)}{i^n} $$ where $i = \sqrt{-1}$ denotes the imaginary unit. A recurrence relation for $p_n$ easily follows:
$$ p_{-1}(x) = 0 \qquad p_0(x) = 1 \qquad p_{n+1}(x) = (x+1) p_n(x) + n^2 p_{n-1}(x) $$