In order to find a formula of the partition of an integer into 5 parts (see $(6)$ below), I find the sequence $$S_1: 3,9,24,21,36,30,75,120,270,462,837,1320,2085,\ldots \tag1$$
It's clear that $S_1$ is divisible by $3$, so I got the sequence $$S_2: 1,3,8,7,12,10,25,40,90,154,279,440,695,\ldots \tag2$$
I can't figure out what's this sequence. I tried to divide the sequence into two other sequences:
The odd sequence $$S_3: 3,7,10,40,154,440,\ldots\tag3$$ and the even sequence $$S_4: 1,8,12,25,90,279,695,\ldots\tag4$$ but yet I can't find any answer.
More details:
For example, the formula of the partition of an integer into 3 parts is A069905
$$p_3(n)=\begin{cases} \frac1{12}n^2 &\text{if}\; n=6k \\[4pt] \frac1{12}(n^2-1) &\text{if}\; n=6k+1 \;\text{or}\; n=6k+5 \\[4pt] \frac1{12}(n^2-4) &\text{if}\; n=6k+2 \;\text{or}\; n=6k+4 \\[4pt] \frac1{12}(n^2+3) &\text{if}\; n=6k+3 \end{cases}\tag5$$
$p_3(n)$ is a mod $6$ formula. I managed to find also $p_4(n)$, and it's a mod $12$ formula, A026810. I suspect that $p_5(n)$ is a mod $15$ formula.
While working in the case $n=15k+5$, I found $$\begin{align} p_5(5) &=\frac13((\phantom{9}5 \cdot\phantom{99}0)+\phantom{9}\color{red}{3})+0 &&=1 \\[4pt] p_5(20)&=\frac13((20\cdot\phantom{9}12)+\phantom{9}\color{red}{9})+1 &&=84 \\[4pt] p_5(35)&=\frac13((35\cdot\phantom{9}57)+\color{red}{24})+1 &&=674 \\[4pt] p_5(50)&=\frac13((50\cdot156)+\color{red}{21})+4 &&=2611 \\[4pt] p_5(65)&=\frac13((65\cdot330)+\color{red}{36})+4 &&=7166 \\[4pt] p_5(80)&=\frac13((80\cdot600)+\color{red}{30})+9 &&=16019 \\[4pt] p_5(95)&=\frac13((95\cdot987)+\color{red}{75})+9 &&=31289\\[4pt] \cdots &= \cdots \end{align} \tag6$$
The sequence of multipliers: $0,12,57,156,330,600,987,\ldots$ is the sequence $3a(n)$, with $a(n) = \frac16n(n + 1)(7n + 5)$.
The sequence: $0,1,1,4,4,9,9,\ldots$ is the squares sequence ($0$ for $0$).
Then the sequence $S_1$ appears highlighted in red.
Any answer please?
The number of partitions of n into 5 parts is https://oeis.org/A026811 which has the generating function $$ x^5/[(1-x)(1-x^2)(1-x^3)/(1-x^4)/(1-x^5)]$$. So the number of ways of paritions of 5n into 5 parts is 1,84,674,2611,.. and has the generating function $$ g_1 = ( -1-81*x-424*x^2-759*x^3-845*x^4-759*x^5-424*x^6-81*x^7-x^8 ) / ( (x^2+1)*(1+x)^2*(x-1)^5 ) $$ Now the generating function of 0,1,1,4,4,9,9 (squares repeated) is $$ g_2= -x*(1+x^2) / ( (1+x)^2*(x-1)^3 ). $$ Subtracting both generating functions and multiplication with 3 gives the generating function for the sequence 53+0, 2012+9, 3557+24,... Since the generating function of 3,12,57,156,... =3,... $1/2*n*(n+1)*(7*n+5)$ is $$3+ 3*x*(4+3*x) ) / (x-1)^4 $$ the generating function of 53, 2012, 3557, 50156 is 15,... 1/2n*(n+1)(7n+5)(15n+5) $$ g_3 = 15 -15*x*(5*x+16)*(3*x+1) / ( (x-1)^5 ) $$ so the generating function for 0,9,24,21,... is (note the mismatch for the 0th term because the $p_5(5)$ value does not match the formula): $$ 3(g_1-g_2)-g_3 -> -3*x*(3-x-11*x^2+13*x^3-8*x^4+13*x^5+7*x^6-2*x^7+x^8)/(x^2+1)/(1+x)^2/(x-1)^5$$ with a recurrence of the type $$a(n)= +3*a(n-1) -2*a(n-2) -2*a(n-3) +4*a(n-4) -4*a(n-5) +2*a(n-6) +2*a(n-7) -3*a(n-8) +a(n-9)$$ for $n-9>0$.