A377198 Expansion of 1/(1 - 4*x/(1-x)^2)^(3/2).

1, 6, 42, 278, 1794, 11382, 71338, 443046, 2732034, 16751462, 102235050, 621535158, 3766261506, 22758222294, 137186860842, 825211984710, 4954574749698, 29697908825286, 177746214414634, 1062416305340502, 6342559258130178, 37823152988963126, 225328426205608362

Offset

0,2

Links

Stefano Spezia, Table of n, a(n) for n = 0..1200

Formula

a(0) = 1; a(n) = 2 * Sum_{k=0..n-1} (3-k/n) * (n-k) * a(k).

a(n) = ((7*n-1)*a(n-1) - (7*n-20)*a(n-2) + (n-3)*a(n-3))/n for n > 2.

a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(n+k-1,n-k).

a(n) ~ 2^(1/4) * sqrt(n) * (1 + sqrt(2))^(2*n) / sqrt(Pi). - Vaclav Kotesovec, May 03 2025

a(n) = 6*n*hypergeom([5/2, 1-n, 1+n], [3/2, 2], -1) for n > 0. - Stefano Spezia, May 08 2025

Mathematica

Table[Sum[(2*k+1)*Binomial[2*k, k]*Binomial[n+k-1, n-k], {k, 0, n}], {n, 0, 35}] (* Vincenzo Librandi, May 11 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, (2*k+1)*binomial(2*k, k)*binomial(n+k-1, n-k));
(Magma) R<x>:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x/(1-x)^2)^(3/2))); // Vincenzo Librandi, May 11 2025

Crossrefs

Cf. A002457, A377197.

Cf. A377194.

Sequence in context: A331706 A074429 A062310 * A229247 A105482 A242158

Adjacent sequences: A377195 A377196 A377197 * A377199 A377200 A377201

Keyword

nonn

Author

Seiichi Manyama, Oct 19 2024

Status

approved