OFFSET
0,2
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
KEYWORD
easy,nonn
AUTHOR
Emanuele Munarini, May 07 2003
STATUS
approved