A378402 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+3*k,n).

1, 5, 49, 533, 6081, 71365, 853105, 10331221, 126321409, 1556044805, 19280812849, 240054516245, 3000687934401, 37634567169477, 473365023172209, 5968699720282453, 75422513665403905, 954877942305602053, 12109526474814388273, 153801537443722913813, 1956059583033416379841

Offset

0,2

Links

Harvey P. Dale, Table of n, a(n) for n = 0..898

Formula

a(n) = [x^n] (1 + x + (1 + x)^4)^n.

a(n) ~ sqrt(1/6 + sqrt(46 - 21/(19 - 3*sqrt(2))^(1/3) - 3*(19 - 3*sqrt(2))^(1/3))/48 + sqrt(23/144 + 7/(192*(19 - 3*sqrt(2))^(1/3)) + (19 - 3*sqrt(2))^(1/3)/192 + 37/(36*sqrt(46 - 21/(19 - 3*sqrt(2))^(1/3) - 3*(19 - 3*sqrt(2))^(1/3))))/2)* (91/27 + 2*sqrt(2*(944 - 81/(1 + sqrt(2))^(1/3) + 81*(1 + sqrt(2))^(1/3)))/27 + sqrt(60416/729 + 32/(9*(1 + sqrt(2))^(1/3)) - 32*(1 + sqrt(2))^(1/3)/9 + 1327808* sqrt(2/(944 - 81/(1 + sqrt(2))^(1/3) + 81*(1 + sqrt(2))^(1/3)))/729)/2)^n / sqrt(Pi*n). - Vaclav Kotesovec, Oct 19 2025

Mathematica

Table[Sum[Binomial[n, k]Binomial[n+3k, n], {k, 0, n}], {n, 0, 20}] (* Harvey P. Dale, Nov 27 2024 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(n+3*k, n));

Crossrefs

Cf. A001850, A243116.

Cf. A378378.

Sequence in context: A212818 A195206 A081474 * A370097 A274671 A371364

Adjacent sequences: A378399 A378400 A378401 * A378403 A378404 A378405

Keyword

nonn

Author

Seiichi Manyama, Nov 25 2024

Status

approved