A294037 -id:A294037 - cp's OEIS frontend

A294035 a(n) = 3^n*hypergeom([-n/3, (1-n)/3, (2-n)/3], [1, 1], -1).

1, 3, 9, 33, 153, 783, 4059, 21087, 110889, 592899, 3214989, 17608077, 97150491, 539331237, 3010588317, 16887545793, 95134584969, 537942476907, 3051902823849, 17365639042449, 99076018204413, 566622950463099, 3247670747106927, 18651711493531539, 107315246617831179

Offset: 0

Peter Luschny, Nov 02 2017

nonn

Diagonal of rational function 1/(1 - (x^3 + y^3 + z^3 + 3*x*y*z)). - Gheorghe Coserea, Aug 04 2018

Robert Israel, Table of n, a(n) for n = 0..1288

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) = this seq., H(4, n, -1) = A294037(n).

T := (m,n,x) -> m^n*hypergeom([seq((k-n)/m, k=0..m-1)], [seq(1,k=0..m-2)], x):
seq(simplify(T(3, n, -1)), n=0..39);

Table[3^n * HypergeometricPFQ[{-n/3, (1 - n)/3, (2 - n)/3}, {1, 1}, -1], {n, 0, 30}] (* Vaclav Kotesovec, Nov 02 2017 *)

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(3, n, -1).
a(n) ~ sqrt(3) * 6^n / (Pi*n) . - Vaclav Kotesovec, Nov 02 2017
-(54*(n+2))*(n+1)*a(n)+27*(n+2)^2*a(n+1)-(3*(3*n^2+15*n+19))*a(n+2)+(n+3)^2*a(n+3) = 0. - Robert Israel, Nov 02 2017

A294036 a(n) = 4^n*hypergeom([-n/4, (1-n)/4, (2-n)/4, (3-n)/4], [1, 1, 1], 1).

Original entry on oeis.org

1, 4, 16, 64, 280, 1504, 9856, 70144, 498136, 3449440, 23506816, 160566784, 1115048896, 7905796864, 56994288640, 414928113664, 3034880623576, 22255957312864, 163667338903936, 1208070406612480, 8955840250934080, 66678657938510080

Offset: 0

Peter Luschny, Nov 02 2017

nonn

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

H(1, n, 1) = A000007(n), H(2, n, 1) = A000984(n), H(3, n, 1) = A006077(n), H(4, n, 1) = this seq., H(1, n, -1) = A000079(n), H(2, n, -1) = A098335(n), H(3, n, -1) = A294035(n), H(4, n, -1) = A294037(n).

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));

Table[4^n * HypergeometricPFQ[{-n/4, (1-n)/4, (2-n)/4, (3-n)/4}, {1, 1, 1}, 1], {n, 0, 30}] (* Vaclav Kotesovec, Nov 02 2017 *)

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).

a(n) ~ 2^(3*n + 2) / (Pi*n)^(3/2). - Vaclav Kotesovec, Nov 02 2017

cp's OEIS Frontend

A294035 a(n) = 3^n*hypergeom([-n/3, (1-n)/3, (2-n)/3], [1, 1], -1).

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