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 A124311 a(n) = Sum_{i=0..n} (-2)^i*binomial(n,i)*B(i) where B(n) = Bell numbers A000110(n). 14
 1, -1, 5, -21, 121, -793, 5917, -49101, 447153, -4421105, 47062773, -535732805, 6484924585, -83079996041, 1121947980173, -15915567647101, 236442490569825, -3668776058118881, 59316847871113445, -997182232031471477, 17397298225094055897, -314449131128077197561 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The sequence has strictly alternating signs. The variant Dobinski-type formula e^(-1)* (2)^n * Sum_{k >= 0} ( (k-1/2)^n / k! ) is strictly positive. - Karol A. Penson and Olivier Gérard, Oct 22 2007 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 FORMULA E.g.f.: exp(exp(-2*x) - 1 + x). - Vladeta Jovovic, Aug 04 2007 G.f.: 1/U(0) where U(k)= 1 + x*(2*k+1) - 4*x^2*(k+1)/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 11 2012 a(n) ~ (-2)^n * n^(n - 1/2) * exp(n/LambertW(n) - n - 1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n - 1/2)). - Vaclav Kotesovec, Jun 26 2022 a(0) = 1; a(n) = a(n-1) + Sum_{k=1..n} binomial(n-1,k-1) * (-2)^k * a(n-k). - Ilya Gutkovskiy, Nov 29 2023 MATHEMATICA Table[ Sum[ (-2)^(k) Binomial[n, k] BellB[k], {k, 0, n}], {n, 0, 50}] (* Karol A. Penson and Olivier Gérard, Oct 22 2007 *) With[{nn=30}, CoefficientList[Series[Exp[Exp[-2x]-1+x], {x, 0, nn}], x] Range[0, nn]!] (* Harvey P. Dale, Mar 04 2016 *) PROG (Sage) def A124311_list(n): # n>=1 T = [0]*(n+1); R = [1] for m in (1..n-1): a, b, c = 1, 0, 0 for k in range(m, -1, -1): r = a + 2*(k*(b+c)+c) if k < m : T[k+2] = u; a, b, c = T[k-1], a, b u = r T[1] = u; R.append((-1)^m*sum(T)) return R A124311_list(22) # Peter Luschny, Nov 02 2012 (SageMath) def A124311(n): return sum( (-2)^k*binomial(n, k)*bell_number(k) for k in range(n+1) ) [A124311(n) for n in range(31)] # G. C. Greubel, Aug 25 2023 (Magma) A124311:= func< n | (&+[(-2)^k*Binomial(n, k)*Bell(k): k in [0..n]]) >; [A124311(n): n in [0..30]]; // G. C. Greubel, Aug 25 2023 CROSSREFS Cf. A000110, A000296, A005493, A126390, A126617. Sequence in context: A168598 A002711 A218962 * A353736 A208593 A213009 Adjacent sequences: A124308 A124309 A124310 * A124312 A124313 A124314 KEYWORD sign AUTHOR N. J. A. Sloane, Aug 04 2007 STATUS approved

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Last modified April 24 00:30 EDT 2024. Contains 371917 sequences. (Running on oeis4.)