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

0,2

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

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

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 *)

Crossrefs

Cf. A000108, A000142, A277175, A277359.

Sequence in context: A125273 A336070 A187761 * A130908 A200404 A000772

Adjacent sequences: A277173 A277174 A277175 * A277177 A277178 A277179

Keyword

nonn

Author

Alois P. Heinz, Oct 02 2016

Status

approved