A327006 a(n) = A327005(n, n).

1, 0, 1, 2, 6, 24, 105, 510, 2765, 16408, 105210, 724580, 5330149, 41649828, 344120777, 2995027126, 27368953170, 261825429024, 2615385871053, 27216432127818, 294443944669341, 3305528914953420, 38442535155671262, 462431164589185924, 5745587267806107545

Offset: 1

Author

Peter Luschny, Aug 13 2019

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 1..540
• Peter Luschny, The Bell transform.
• Retoz, Sous-dérangements, Mai 2021.

Crossrefs

Cf. A005387, A264428, A327005.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!(Laplace( Exp((x-1)*Exp(x) + 1) ))); // G. C. Greubel, Nov 17 2022

Maple

# BellMatrix is defined in A264428.
a := proc(n) BellMatrix(j -> modp(j, n), n): add(i, i in %[n]) end:
seq(a(n), n=1..25);
# Or, assuming offset = 0:
ser := series(exp(1 + (x - 1)*exp(x)), x=0, 25):
seq(n!*coeff(ser, x, n), n = 0..24); # Contributed by C. L. Martin (Retoz), May 16 2021.

Mathematica

With[{m=40}, CoefficientList[Series[Exp[(x-1)*Exp[x] +1], {x,0,m}], x]*Range[0, m]!] (* G. C. Greubel, Nov 17 2022 *)

SageMath

def A327006_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( exp((x-1)*exp(x) +1) ).egf_to_ogf().list()
A327006_list(40) # G. C. Greubel, Nov 17 2022

Formula

a(n) = Sum_{k=1..n} BM[n][k] where BM is the BellMatrix(j -> j mod n) as defined in A264428.

Assuming offset = 0:

a(n) = n! * [x^n] exp(1 + (x - 1)*exp(x)). - Contributed by C. L. Martin (Retoz), May 16 2021.

a(n) = Sum_{j=0..n} binomial(n, j) * (-2)^(n-j) * A005387(j). - G. C. Greubel, Nov 17 2022