A277175 Convolution of Catalan numbers and factorial numbers.

1, 2, 5, 15, 53, 222, 1120, 6849, 50111, 427510, 4142900, 44693782, 529276962, 6813205468, 94642629984, 1410507388421, 22445134308123, 379776665469030, 6808016435182620, 128886547350655050, 2569493300908367550, 53805226930896987540, 1180673761078007109840

Offset: 0

Author

Alois P. Heinz, Oct 02 2016

Keywords

• nonn

Links

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

Crossrefs

Cf. A000108, A000142, A277176, A277359, A277396.

Programs

Maple

a:= proc(n) option remember; `if`(n<4, [1, 2, 5, 15][n+1],
      ((2*(n^4-n^3-19*n^2+48*n-5))*a(n-1)
       -(n+1)*(n^4+9*n^3-90*n^2+226*n-160)*a(n-2)
       +(2*(4*n^5-18*n^4-23*n^3+266*n^2-523*n+330))*a(n-3)
       -(4*(n-2))*(n^2-4*n+5)*(2*n-5)^2*a(n-4))/
       ((n+1)*(n^2-6*n+10)))
    end:
seq(a(n), n=0..30);

Mathematica

Table[Sum[CatalanNumber[k]*(n - k)!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Oct 13 2016 *)

Formula

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

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