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

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

Offset: 0

Author

Ilya Gutkovskiy, Apr 08 2025

Keywords

• nonn

Comments

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

Crossrefs

Cf. A001850, A005259, A069835, A382405.

Programs

Mathematica

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}]

Formula

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