A382849 a(n) = Sum_{k=0..n} (-1)^(n-k) * (binomial(n,k) * binomial(n+k,k))^2.

1, 3, 1, -357, -6999, -62997, 444529, 27783003, 508019689, 3206511003, -89889084999, -3274278527517, -49395223500999, -66079827133317, 16197028704290001, 433384098559415643, 4988878584849669609, -35687369703800052357, -2815548294132454060151, -58942279760573467233357

Offset: 0

Author

Ilya Gutkovskiy, Apr 06 2025

Keywords

• sign

Comments

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

Crossrefs

Cf. A005258, A005259, A126869, A176335, A228304, A382848.

Programs

Mathematica

Table[Sum[(-1)^(n - k) (Binomial[n, k] Binomial[n + k, k])^2, {k, 0, n}], {n, 0, 19}]
Table[(-1)^n HypergeometricPFQ[{-n, -n, n + 1, n + 1}, {1, 1, 1}, -1], {n, 0, 19}]
Table[SeriesCoefficient[1/(1 - y + z + x y + z w + x z + x y w + x y z w), {x, 0, n}, {y, 0, n}, {z, 0, n}, {w, 0, n}], {n, 0, 19}]