A213170 E.g.f.: exp(2*(1-exp(x))).

1, -2, 2, 2, -6, -14, 26, 178, 90, -2382, -9446, 13746, 287194, 998578, -3687782, -56264782, -208446118, 1017677490, 17194912282, 79540574642, -317691584294, -7577787031374, -47958156443238, 77252406086578, 4400217218583642, 39757699729476274, 54538870133137690

Offset: 0

Author

Philippe Deléham, Feb 14 2013

Keywords

• sign
• easy

Comments

Exponential self-convolution of complementary Bell numbers (A000587). - Vladimir Reshetnikov, Oct 07 2016

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..580
• Vaclav Kotesovec, Graph - asymptotic (1000 terms)

Crossrefs

Cf. A000587, A001861, A007318, A048993.

Programs

Maple

b:= proc(n, m) option remember; `if`(n=0,
      (-2)^m, m*b(n-1, m)+b(n-1, m+1))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..27);  # Alois P. Heinz, Jul 17 2022

Mathematica

CoefficientList[Series[E^(2*(1-E^x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Aug 04 2014 *)
Table[BellB[n, -2], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

PARI

x='x+O('x^50); Vec(serlaplace(exp(2*(1-exp(x))))) \\ G. C. Greubel, Nov 15 2017

Formula

a(n) = Sum_{k=0..n} A048993(n,k)*(-2)^k.

a(n) = Sum_{k=0..n} A000587(k)*A000587(n-k)*binomial(n,k).

G.f.: 1/(1+2*x/(1-x/(1+2*x/(1-2*x/(1+2*x/(1-3*x/(1+2*x/(1-4*x/(1+2*x/(1-...(continued fraction).

Sum_{k=0..n} binomial(n,k)*a(k) = a(n+1)/(-2). - Philippe Deléham, Feb 17 2013

G.f.: 1/Q(0) where Q(k) = 1 + x*(k+1) + x/(1 - 2*x*(k+1)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 07 2013

Lim sup n->infinity (abs(a(n))/n!)^(1/n) / abs(exp(1/LambertW(-n/2)) / LambertW(-n/2)) = 1. - Vaclav Kotesovec, Aug 04 2014

a(n) = B_n(-2), where B_n(x) is n-th Bell polynomial. - Vladimir Reshetnikov, Oct 20 2015

G.f.: Sum_{j>=0} (-2)^j*x^j / Product_{k=1..j} (1 - k*x). - Ilya Gutkovskiy, Apr 06 2019