A294037 a(n) = 4^n*hypergeom([-n/4, (1-n)/4, (2-n)/4, (3-n)/4], [1, 1, 1], -1).

1, 4, 16, 64, 232, 544, -1664, -37376, -362024, -2743712, -17780864, -98955776, -442825664, -1129423616, 5536033792, 118591811584, 1224814969816, 9905491019104, 68032143081856, 398051159254528, 1854461906222272, 4784426026102528

Offset

0,2

Comments

Diagonal of rational function 1/(1 - (x^4 + y^4 + z^4 - t^4 + 4*x*y*z*t)). - Gheorghe Coserea, Aug 04 2018

Links

Table of n, a(n) for n=0..21.

Formula

Let H(m, n, x) = m^n*hypergeom([(k-n)/m for k=0..m-1], [1 for k=0..m-2], x) then a(n) = H(4, n, -1).

From Peter Bala, Sep 26 2025: (Start)

a(n) = Sum_{k = 0..floor(n/4)} (-1)^k * 4^(n-4*k) * n!/((n-4*k)!*k!^4).

E.g.f.: exp(4*x) * Sum_{k >= 0} (-1)^k * x^(4*k)/k!^4 = 1 + 4*x + 16*x^2/2! + 64*x^3/3! + 232*x^4/4! + 544*x^5/5! - ....

n^3*a(n) = 4*(2*n - 1)*(2*n^2 - 2*n + 1)*a(n-1) - 16*(n - 1)*(6*n^2 - 12*n + 7)*a(n-2) + 128*(n - 1)*(n - 2)*(2*n - 3)*a(n-3) - 512*(n - 1)*(n - 2)*(n - 3)*a(n-4) with a(0) = 1, a(1) = 4, a(2) = 16 and a(3) = 64. (End)

Maple

T := (m, n, x) -> m^n*hypergeom([seq((k-n)/m, k=0..m-1)], [seq(1, k=0..m-2)], x):
lprint(seq(simplify(T(4, n, -1)), n=0..39));

Mathematica

Table[4^n * HypergeometricPFQ[{-n/4, (1-n)/4, (2-n)/4, (3-n)/4}, {1, 1, 1}, -1], {n, 0, 20}] (* Vaclav Kotesovec, Nov 02 2017 *)

Crossrefs

H(1, n, 1) = A000007(n), H(2, n, 1) = A000984(n), H(3, n, 1) = A006077(n), H(4, n, 1) = A294036(n), H(1, n, -1) = A000079(n), H(2, n, -1) = A098335(n), H(3, n, -1) = A294035(n), H(4, n, -1) = this seq..

Sequence in context: A065738 A248089 A248088 * A228735 A289694 A232425

Adjacent sequences: A294034 A294035 A294036 * A294038 A294039 A294040

Keyword

sign,easy

Author

Peter Luschny, Nov 02 2017

Status

approved