A277176 Exponential convolution of Catalan numbers and factorial numbers.

1, 2, 6, 23, 106, 572, 3564, 25377, 204446, 1844876, 18465556, 203179902, 2438366836, 31699511768, 443795839192, 6656947282725, 106511191881270, 1810690391626380, 32592427526913540, 619256124778620450, 12385122502136529420, 260087572569333384840

Offset: 0

Author

Alois P. Heinz, Oct 02 2016

Keywords

• nonn

Comments

a(n) = number of permutations of [n+1] in which the first entry does not start a (classical) 1234 pattern. The number of such permutations with first entry i is n!/(n + 1 - i)! C(n + 1 - i) where C(n) is the Catalan number A000108(n). - David Callan, Jun 12 2017

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..449

Crossrefs

Cf. A000108, A000142, A277175, A277359.

Programs

Maple

a:= proc(n) option remember; `if`(n<2, n+1,
     ((n^2+5*n-2)*a(n-1)-(4*n-2)*(n-1)*a(n-2))/(n+1))
    end:
seq(a(n), n=0..30);

Mathematica

a[n_] := Sum[Binomial[n, i] CatalanNumber[i] (n-i)!, {i, 0, n}];
a /@ Range[0, 30] (* Jean-François Alcover, Dec 21 2020 *)

Formula

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

a(n) = Sum_{i=0..n} binomial(n,i) * C(i) * (n-i)!.

a(n) ~ exp(2) * BesselI(2,2) * n!. - Vaclav Kotesovec, Oct 13 2016