A299507 a(n) = (-1)^n*hypergeom([-n, n], [1], 4).

1, 3, 33, 387, 4737, 59523, 761121, 9854211, 128772609, 1694927619, 22437369633, 298419470979, 3984500221569, 53376363001731, 717044895641121, 9656091923587587, 130310873022310401, 1761872309456567811, 23861153881099854369, 323634591584064809859

Offset

0,2

Links

Table of n, a(n) for n=0..19.

Formula

From Vaclav Kotesovec, Jul 05 2018: (Start)

Recurrence: n*(2*n-3)*a(n) = 2*(14*n^2 - 28*n + 11)*a(n-1) - (n-2)*(2*n-1)*a(n-2).

a(n) ~ 2^(-3/2) * 3^(1/4) * (7 + 4*sqrt(3))^n / sqrt(Pi*n). (End)

From Peter Bala, April 18 2024: (Start)

a(n) = Sum_{k = 0..n} binomial(n, k)*binomial(n+k-1, k-1)*3^k = R(n, 3) for n >= 1, where R(n, x) denotes the n-th row polynomial of A253283.

a(n) = 3*n* hypergeom([1 - n, n + 1], [2], -3) for n >= 1.

a(n) = (1/2)*(LegendreP(n, 7) - LegendreP(n-1, 7)) for n >= 1.

a(n) = [x^n] ( (1 - x)/(1 - 4*x) )^n.

It follows that the Gauss congruences hold: a(n*p^r) == a(n*p^(r-1)) (mod p^r) for all primes p and positive integers n and r.

G.f.: (sqrt(x^2 - 14*x + 1) - x + 1)/(2*sqrt(x^2 - 14*x + 1)) = 1 + 3*x + 33*x^2 + 387*x^3 + .... (End)

Maple

seq(simplify( (-1)^n*hypergeom([-n, n], [1], 4)), n = 0..20); # Peter Bala, Apr 18 2024

Mathematica

a[n_] := (-1)^n Hypergeometric2F1[-n, n, 1, 4]; Table[a[n], {n, 0, 19}]

Crossrefs

Cf. A299864, A245926, A084768, A245927, A253283.

Cf. A300945, A300946.

Sequence in context: A228002 A163476 A155660 * A294167 A009502 A222941

Adjacent sequences: A299504 A299505 A299506 * A299508 A299509 A299510

Keyword

nonn

Author

Peter Luschny, Mar 16 2018

Status

approved