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A277176 Exponential convolution of Catalan numbers and factorial numbers.

%I A277176 #18 Dec 21 2020 07:16:29
%S A277176 1,2,6,23,106,572,3564,25377,204446,1844876,18465556,203179902,
%T A277176 2438366836,31699511768,443795839192,6656947282725,106511191881270,
%U A277176 1810690391626380,32592427526913540,619256124778620450,12385122502136529420,260087572569333384840
%N A277176 Exponential convolution of Catalan numbers and factorial numbers.
%C A277176 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
%H A277176 Alois P. Heinz, <a href="/A277176/b277176.txt">Table of n, a(n) for n = 0..449</a>
%F A277176 E.g.f.: exp(2*x)/(1-x)*(BesselI(0,2*x)-BesselI(1,2*x)).
%F A277176 a(n) = Sum_{i=0..n} binomial(n,i) * C(i) * (n-i)!.
%F A277176 a(n) ~ exp(2) * BesselI(2,2) * n!. - _Vaclav Kotesovec_, Oct 13 2016
%p A277176 a:= proc(n) option remember; `if`(n<2, n+1,
%p A277176      ((n^2+5*n-2)*a(n-1)-(4*n-2)*(n-1)*a(n-2))/(n+1))
%p A277176     end:
%p A277176 seq(a(n), n=0..30);
%t A277176 a[n_] := Sum[Binomial[n, i] CatalanNumber[i] (n-i)!, {i, 0, n}];
%t A277176 a /@ Range[0, 30] (* _Jean-François Alcover_, Dec 21 2020 *)
%Y A277176 Cf. A000108, A000142, A277175, A277359.
%K A277176 nonn
%O A277176 0,2
%A A277176 _Alois P. Heinz_, Oct 02 2016