Question

If $(1+px+x^2)^n=1+a_{1}x+a_{2} x^2+....+a_{2n}x^{2n}$, then prove that $(np-pr)a_{r}=(r+1)a_{r+1}+(r-1-2n)a_{r-1}$ for $1<r<2n$

score 78 · Answer 1 · 2026-03-26 01:04:48

78

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If $(1+px+x^2)^n=1+a_{1}x+a_{2} x^2+....+a_{2n}x^{2n}$, then prove that $(np-pr)a_{r}=(r+1)a_{r+1}+(r-1-2n)a_{r-1}$ for $1<r<2n$

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score 120 · Answer 2 · 2026-04-12 01:45:57

120

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54

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score 107 · Answer 4 · 2026-04-12 08:01:09

107

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score 195 · Answer 5 · 2026-04-16 19:50:36

195

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45

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score 869 · Answer 10 · 2026-04-17 14:06:05

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score 109 · Answer 11 · 2026-04-16 01:28:33

109

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score 121 · Answer 12 · 2026-04-11 01:29:46

121

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score 132 · Answer 14 · 2026-04-12 16:41:57

132

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score 68 · Answer 15 · 2026-04-15 22:29:53

68

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List Question

If $(1+px+x^2)^n=1+a_{1}x+a_{2} x^2+....+a_{2n}x^{2n}$, then prove that $(np-pr)a_{r}=(r+1)a_{r+1}+(r-1-2n)a_{r-1}$ for $1<r<2n$

Estimates of the sum of the biggest terms in a binomial expansion.

Does the result given below work for composite numbers?

Are there any numerically-plausible perfect binary code parameters besides (90,2)?

Conjecture: $\binom{n}{k } \mod m =0$ for all $k=1,2,3,\dots,n-1$ only when $m $ is a prime number and $n$ is a power of $m$

Is there such a known function?

How do the graphs of the "Birthday Probability" functions work?

Coefficient of polynomial of $x^k$

Combinatorial Identity from $\textrm{sinc}(x)$ integrals

Why is the binomial coefficient $\binom{\text{even}}{\text{odd}}$ always even?

Finding $ \sum_{k=0}^n (-1)^k A_k {n \choose k}, ~\text{if}~ (1+x+x^2)^n=\sum_{k=0}^{2n} A_k x^k$

Binomial Theorem proof using three terms

Equivalence of two binomial sums

Enumerating $m\times n$ binary matrices with constant column sum and at least one non-zero row entry

Prove number of single color triangles in complete graph

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