A382848 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k)^2 * binomial(n+k,k).

1, 1, -5, -35, -29, 751, 3991, -4115, -137885, -495269, 2114245, 25786795, 50109775, -627370925, -4643568305, -495798035, 157753390435, 768269873875, -1851203127335, -35924154988865, -107001450483779, 763444753890721, 7510024190977105, 8899910747771995

Offset

0,3

Comments

Diagonal of the rational function 1 / (1 + x + x*y + y*z + x*z + x*y*z).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Formula

(59*n-94)*n^2*a(n) = 5*(59*n^3-153*n^2+117*n-30)*a(n-1) - (2301*n^3-8268*n^2+9257*n-3050)*a(n-2) - 2*(59*n-35)*(n-2)^2*a(n-3) with a(0) = 1, a(1) = 1 and a(2) = -5. - Peter Bala, May 24 2025

From Seiichi Manyama, Feb 16 2026: (Start)

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

a(n) = Sum_{k=0..n} (-1)^k * binomial(n+k,k) * Sum_{j=0..n-k} binomial(n+j,j) * multinomial(n; j, k, n-j-k). (End)

Mathematica

Table[Sum[(-1)^(n - k) Binomial[n, k]^2 Binomial[n + k, k], {k, 0, n}], {n, 0, 23}]
Table[(-1)^n HypergeometricPFQ[{-n, -n, n + 1}, {1, 1}, -1], {n, 0, 23}]
Table[SeriesCoefficient[1/(1 + x + x y + y z + x z + x y z), {x, 0, n}, {y, 0, n}, {z, 0, n}], {n, 0, 23}]

Crossrefs

Cf. A005258, A026641, A126869, A245086, A274666, A274674, A382405, A382849.

Sequence in context: A087675 A376610 A128044 * A299529 A014632 A117985

Adjacent sequences: A382845 A382846 A382847 * A382849 A382850 A382851

Keyword

sign

Author

Ilya Gutkovskiy, Apr 06 2025

Status

approved