A382934 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * binomial(n+2*k,k) * 2^(n-k).

1, 8, 142, 3188, 79306, 2091128, 57251944, 1609275536, 46123258714, 1341870616928, 39505611952852, 1174352843125976, 35189447673190864, 1061579548438995776, 32210037668484980992, 982173609216589910528, 30079350892561552670554, 924711257106480733093616, 28524228913983070512002044

Offset: 0

Author

Ilya Gutkovskiy, Apr 09 2025

Keywords

• nonn

Comments

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

Crossrefs

Cf. A001850, A069835, A081798, A126086.

Programs

Mathematica

Table[Sum[Binomial[n, k] Binomial[n + k, k] Binomial[n + 2 k, k] 2^(n - k), {k, 0, n}], {n, 0, 18}]
Table[2^n HypergeometricPFQ[{n/2 + 1/2, n/2 + 1, -n}, {1, 1}, -2], {n, 0, 18}]
Table[SeriesCoefficient[1/(1 - x - y - z - 2 x y z), {x, 0, n}, {y, 0, n}, {z, 0, n}], {n, 0, 18}]

Formula

a(n) ~ sqrt(1/4 + sqrt(13)*cosh(arccosh(47/13^(3/2))/3)/6) * (1 + 2*cosh(arccosh(2)/3))^(3*n) / (Pi*n). Equivalently, a(n) ~ (1 + (2 - sqrt(3))^(1/3) + (2 + sqrt(3))^(1/3))^(3*n) / (sqrt(6*((2 - sqrt(3))^(1/3) + (2 + sqrt(3))^(1/3) - 2))*Pi*n). - Vaclav Kotesovec, Apr 17 2025