A335309 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * n^(n-k).

1, 3, 22, 245, 3606, 65527, 1411404, 35066313, 985483270, 30869546411, 1065442493556, 40144438269949, 1638733865336764, 72012798200637855, 3388250516614331416, 169894851136173584145, 9041936334960057699654, 508945841697238471315027, 30202327515992972576218980

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..381

Formula

a(n) = central coefficient of (1 + (n + 2)*x + (n + 1)*x^2)^n.

a(n) = [x^n] 1 / sqrt(1 - 2*(n + 2)*x + n^2*x^2).

a(n) = n! * [x^n] exp((n + 2)*x) * BesselI(0,2*sqrt(n + 1)*x).

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

a(n) ~ exp(2*sqrt(n)) * n^(n - 1/4) / (2*sqrt(Pi)) * (1 + 11/(12*sqrt(n))). - Vaclav Kotesovec, Jan 09 2023

Mathematica

Join[{1}, Table[Sum[Binomial[n, k] Binomial[n + k, k] n^(n - k), {k, 0, n}], {n, 1, 18}]]
Table[SeriesCoefficient[1/Sqrt[1 - 2 (n + 2) x + n^2 x^2], {x, 0, n}], {n, 0, 18}]
Table[n! SeriesCoefficient[Exp[(n + 2) x] BesselI[0, 2 Sqrt[n + 1] x], {x, 0, n}], {n, 0, 18}]
Table[Hypergeometric2F1[-n, -n, 1, 1 + n], {n, 0, 18}]

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)^2*(n+1)^k); \\ Michel Marcus, Jun 01 2020

Crossrefs

Cf. A001850, A069835, A084771, A084772, A098659, A187021, A307883, A331656, A335310.

Sequence in context: A376564 A141356 A376558 * A162633 A052677 A136741

Adjacent sequences: A335306 A335307 A335308 * A335310 A335311 A335312

Keyword

nonn

Author

Ilya Gutkovskiy, May 31 2020

Status

approved