A051782 Apply the "Stirling-Bernoulli transform" to Catalan numbers.

1, 0, 2, -12, 122, -1620, 26882, -536172, 12506762, -334261380, 10075002962, -338180323932, 12512502202202, -505992958647540, 22204726014875042, -1050993549782729292, 53373431773793542442, -2894886293042487680100, 167021024758368026331122

Offset: 0

Author

N. J. A. Sloane, Dec 09 1999

Keywords

• sign

Comments

The "Stirling-Bernoulli transform" maps a sequence b_0, b_1, b_2, ... to a sequence c_0, c_1, c_2, ..., where if B has o.g.f. B(x), c has e.g.f. exp(x)*B(1-exp(x)). More explicitly, c_n = Sum_{m=0..n} (-1)^m*m!*Stirling2(n+1,m+1)*b_m.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A000108, A163626.

Programs

Maple

a:= n-> add((-1)^k *k! *Stirling2(n+1, k+1)*binomial(2*k, k)/
        (k+1), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, May 17 2013

Mathematica

a[n_] := Sum[(-1)^k k! StirlingS2[n+1, k+1] CatalanNumber[k], {k, 0, n}];
Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Apr 06 2016 *)

Formula

a(n) = Sum_{k = 0..n} A163626(n,k)*A000108(k). - Philippe Deléham, May 25 2015