A217946 a(n) = 4^n*(n+1)*(8*n^2+32*n+33)*P(3/2,n)/(3*P(4,n)) where P(a,n) is the Pochhammer rising factorial.

11, 73, 387, 1876, 8670, 38907, 171171, 742456, 3186378, 13562770, 57352526, 241234488, 1010195420, 4214583135, 17527709475, 72695369520, 300782736210, 1241908383870, 5118246664410, 21058891783800, 86518038936420, 354975217564110, 1454668818567822, 5954594437631376, 24350248227272100, 99484144007729572

Offset

0,1

Links

Robert Israel, Table of n, a(n) for n = 0..1654

Ping Sun, Proof of two conjectures of Petkovsek and Wilf on Gessel walks Discrete Math. 312, No. 24 (2012), 3649-3655. MR2979494. See Th. 1.1, case 3.

Formula

From Robert Israel, Mar 28 2018: (Start)

(n+1)^2*(n+4)*(8*n^2+32*n+33)*a(n+1) = 2*(2*n+3)*(n+2)*(8*n^2+48*n+73)*a(n).

G.f.: (3-x)/(2*x^3) - (3-19*x+24*x^2-16*x^3)/(2*(1-4*x)^(3/2)*x^3). (End)

a(n) ~ 2^(2*n+5) * sqrt(n/Pi). - Amiram Eldar, Sep 01 2025

Maple

f:= n -> 4^n*(n+1)*(8*n^2+32*n+33)*pochhammer(3/2, n)/(3*pochhammer(4, n)):
map(f, [$0..40]); # Robert Israel, Mar 28 2018

Mathematica

a[n_] := 4^n * (n+1) * (8*n^2 + 32*n + 33) * Pochhammer[3/2, n] / (3 * Pochhammer[4, n]); Array[a, 26, 0] (* Amiram Eldar, Sep 01 2025 *)

Crossrefs

Sequence in context: A142015 A123039 A226034 * A163775 A092244 A342830

Adjacent sequences: A217943 A217944 A217945 * A217947 A217948 A217949

Keyword

nonn

Author

N. J. A. Sloane, Nov 07 2012

Status

approved