A265023 Second order complementary Bell numbers.

1, -1, 2, -4, 9, -22, 54, -139, 372, -948, 2607, -7388, 16058, -58957, 174854, 210448, 4345025, -2008714, -165872030, -1756557123, -6144936528, 60244093040, 1164910003567, 8228177887688, -10562519450714, -967088274083133, -11322641425582454, -37483806372774364

Offset: 0

Author

Peter Luschny, Dec 03 2015

Keywords

• sign

Crossrefs

Cf. A000587 (complementary Bell numbers), A264428.

Programs

Mathematica

nmax = 27;
A = Exp[x] + O[x]^(nmax - 1);
B = Exp[1 - Integrate[A, x]]/E;
c = Exp[1 - Integrate[B, x]]/E;
CoefficientList[c, x] Range[0, nmax]! (* Jean-François Alcover, Jul 12 2019, from PARI *)

PARI

\\ For n>28 precision has to be adapted as needed!
A = exp('x + O('x^33) );
B = exp(1 - intformal(A) )/exp(1);
C = exp(1 - intformal(B) )/exp(1);
round(Vec(serlaplace(C)))

Sage

# uses[bell_transform from A264428]
def A265023_list(len):
    uno = [1]*len
    complementary_bell_numbers = [sum((-1)^n*b for (n, b) in enumerate (bell_transform(n, uno))) for n in range(len)]
    complementary_bell_numbers2 = [sum((-1)^n*b for (n, b) in enumerate (bell_transform(n, complementary_bell_numbers))) for n in range(len)]
    return complementary_bell_numbers2
print(A265023_list(28))