A082580 A sum of Lah numbers and binomial coefficients.

1, 2, 10, 68, 574, 5732, 65724, 847800, 12120966, 189890588, 3230531356, 59246895512, 1164225730540, 24387062160008, 542155626123544, 12743158072837680, 315624979700257350, 8213507146488950700

Offset

0,2

Links

Table of n, a(n) for n=0..17.

Formula

Formula: a(n) = Sum[ Lah[ n, k ] binomial[ 2 k, k ], { k, 0, n } ] where Lah[n, k] = binomial[ n - 1, k - 1 ] n! / k! are the Lah numbers. Recurrence: ( n + 3 ) a(n+3) - ( 3 n^2 + 17 n + 24 ) a(n+2) + 3 ( n + 3 )( n + 2 )( n + 1 ) a(n+1) - ( n + 2 )( n + 1 )^2 n a(n) = 0

E.g.f.: BesselI(0, 2*x/(1-x))*exp(2*x/(1-x)). - Vladeta Jovovic, Sep 13 2003

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

Mathematica

RecurrenceTable[{-(-3 + n) (-2 + n)^2 (-1 + n) a[-3 + n] + 3 (-2 + n) (-1 + n) n a[-2 + n] - n (-1 + 3 n) a[-1 + n] + n a[n] == 0, a[0] == 1, a[1] == 2, a[2] == 10}, a, {n, 0, 20}] (* Vaclav Kotesovec, Jun 07 2019 *)

Crossrefs

Sequence in context: A136633 A335729 A361769 * A231492 A136658 A329475

Adjacent sequences: A082577 A082578 A082579 * A082581 A082582 A082583

Keyword

easy,nonn

Author

Emanuele Munarini, May 07 2003

Status

approved