A317276 a(n) = Sum_{k=0..n} binomial(n-1,k-1)*binomial(2*k,k)*n!/(k + 1)!.

1, 1, 4, 23, 170, 1522, 15912, 189513, 2525966, 37176014, 597852056, 10417551806, 195334043764, 3918512356228, 83688324997136, 1894856645139765, 45317092619635350, 1141097574390542550, 30166154721201845400, 835120134797808510690, 24155626083101758391820, 728505545127602209546620

Offset

0,3

Comments

Lah transform of the Catalan numbers (A000108).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..435

N. J. A. Sloane, Transforms

Formula

E.g.f.: exp(2*x/(1 - x))*(BesselI(0,2*x/(1 - x)) - BesselI(1,2*x/(1 - x))).

a(n) ~ exp(4*sqrt(n) - n - 2) * n^(n-1) / (2*sqrt(2*Pi)). - Vaclav Kotesovec, Jun 07 2019

Mathematica

Table[Sum[Binomial[n - 1, k - 1] Binomial[2 k, k] n!/(k + 1)!, {k, 0, n}], {n, 0, 21}]
nmax = 21; CoefficientList[Series[Exp[2 x/(1 - x)] (BesselI[0, 2 x/(1 - x)] - BesselI[1, 2 x/(1 - x)]), {x, 0, nmax}], x] Range[0, nmax]!
Join[{1}, Table[n! HypergeometricPFQ[{3/2, 1 - n}, {2, 3}, -4], {n, 21}]]

Crossrefs

Cf. A000108, A082580.

Sequence in context: A053525 A277382 A208676 * A113869 A084357 A360868

Adjacent sequences: A317273 A317274 A317275 * A317277 A317278 A317279

Keyword

nonn

Author

Ilya Gutkovskiy, Jul 25 2018

Status

approved