A382405 -id:A382405 - cp's OEIS frontend

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

Ilya Gutkovskiy, Apr 06 2025

sign

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

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

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}]

(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

A382642 a(n) = Sum_{k=0..n} (binomial(n,k) * binomial(n+k,k))^2 * 2^(n-k).

Original entry on oeis.org

1, 6, 112, 2784, 79716, 2478936, 81369856, 2774798592, 97345792804, 3490750940376, 127377525333312, 4714499194430592, 176563416839871504, 6678628406445775968, 254781841509308692992, 9791397137378344986624, 378713818451270226094884, 14731112080159997036570328

Offset: 0

Ilya Gutkovskiy, Apr 08 2025

nonn

Diagonal of the rational function 1 / (1 - y - z - x*y - z*w - 2*x*z - x*y*w - 2*x*y*z*w).

Cf. A001850, A005259, A069835, A382405.

Table[Sum[(Binomial[n, k] Binomial[n + k, k])^2 2^(n - k), {k, 0, n}], {n, 0, 17}]
Table[2^n HypergeometricPFQ[{-n, -n, n + 1, n + 1}, {1, 1, 1}, 1/2], {n, 0, 17}]
Table[SeriesCoefficient[1/(1 - y - z - x y - z w - 2 x z - x y w - 2 x y z w), {x, 0, n}, {y, 0, n}, {z, 0, n}, {w, 0, n}], {n, 0, 17}]

a(n) ~ sqrt(6 + 4*sqrt(2) + sqrt(137/2 + 97/sqrt(2))) * (10 + 8*sqrt(2) + 4*sqrt(14 + 10*sqrt(2)))^n / (4 * Pi^(3/2) * n^(3/2)). - Vaclav Kotesovec, Apr 08 2025

cp's OEIS Frontend

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

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