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A277175 Convolution of Catalan numbers and factorial numbers.

%I A277175 #15 Oct 13 2016 03:57:03
%S A277175 1,2,5,15,53,222,1120,6849,50111,427510,4142900,44693782,529276962,
%T A277175 6813205468,94642629984,1410507388421,22445134308123,379776665469030,
%U A277175 6808016435182620,128886547350655050,2569493300908367550,53805226930896987540,1180673761078007109840
%N A277175 Convolution of Catalan numbers and factorial numbers.
%H A277175 Alois P. Heinz, <a href="/A277175/b277175.txt">Table of n, a(n) for n = 0..449</a>
%F A277175 a(n) = Sum_{i=0..n} C(i) * (n-i)!.
%F A277175 a(n) ~ n! * (1 + 1/n + 2/n^2 + 7/n^3 + 31/n^4 + 163/n^5 + 979/n^6 + 6556/n^7 + 48150/n^8 + 383219/n^9 + 3275121/n^10 + ...), for coefficients see A277396. - _Vaclav Kotesovec_, Oct 13 2016
%p A277175 a:= proc(n) option remember; `if`(n<4, [1, 2, 5, 15][n+1],
%p A277175       ((2*(n^4-n^3-19*n^2+48*n-5))*a(n-1)
%p A277175        -(n+1)*(n^4+9*n^3-90*n^2+226*n-160)*a(n-2)
%p A277175        +(2*(4*n^5-18*n^4-23*n^3+266*n^2-523*n+330))*a(n-3)
%p A277175        -(4*(n-2))*(n^2-4*n+5)*(2*n-5)^2*a(n-4))/
%p A277175        ((n+1)*(n^2-6*n+10)))
%p A277175     end:
%p A277175 seq(a(n), n=0..30);
%t A277175 Table[Sum[CatalanNumber[k]*(n - k)!, {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Oct 13 2016 *)
%Y A277175 Cf. A000108, A000142, A277176, A277359, A277396.
%K A277175 nonn
%O A277175 0,2
%A A277175 _Alois P. Heinz_, Oct 02 2016