A374511 Expansion of 1/(1 - 4*x - 4*x^2)^(5/2).

1, 10, 80, 560, 3640, 22512, 134400, 781440, 4451040, 24939200, 137865728, 753625600, 4080643840, 21916106240, 116877312000, 619457482752, 3265293719040, 17128725519360, 89462514606080, 465434423336960, 2412895587536896, 12468681310412800, 64242981906022400

Offset

0,2

Links

Table of n, a(n) for n=0..22.

Formula

a(0) = 1, a(1) = 10; a(n) = (2*(2*n+3)*a(n-1) + 4*(n+3)*a(n-2))/n.

a(n) = (binomial(n+4,2)/6) * Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+2,n-2*k) * binomial(2*k+2,k).

a(n) = 2^(n-3)*Pochhammer(n+1, 4)*hypergeom([(1-n)/2, -n/2], [3], 2)/3. - Stefano Spezia, Jul 10 2024

a(n) = Sum_{k=0..n} (-4)^k * binomial(-5/2,k) * binomial(k,n-k). - Seiichi Manyama, Oct 19 2024

Mathematica

a[n_]:=2^(n-3) Pochhammer[n+1, 4]*Hypergeometric2F1[(1-n)/2, -n/2, 3, 2]/3; Array[a, 23, 0] (* Stefano Spezia, Jul 10 2024 *)

Prog

(PARI) a(n) = binomial(n+4, 2)/6*sum(k=0, n\2, 2^(n-k)*binomial(n+2, n-2*k)*binomial(2*k+2, k));

Crossrefs

Cf. A006139, A374497, A374513.

Sequence in context: A220485 A055285 A036070 * A377199 A257286 A125373

Adjacent sequences: A374508 A374509 A374510 * A374512 A374513 A374514

Keyword

nonn

Author

Seiichi Manyama, Jul 09 2024

Status

approved