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 A089677 Exponential convolution of A000670(n), with A000670(0)=0, with the sequence of all ones alternating in sign. 4
 0, 1, 1, 7, 37, 271, 2341, 23647, 272917, 3543631, 51123781, 811316287, 14045783797, 263429174191, 5320671485221, 115141595488927, 2657827340990677, 65185383514567951, 1692767331628422661, 46400793659664205567, 1338843898122192101557 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS Stirling transform of A005212(n)=[1,0,6,0,120,0,5040,...] is a(n)=[1,1,7,37,271,...]. - Michael Somos, Mar 04 2004 Occurs also as first column of a matrix-inversion occurring in a sum-of-like-powers problem. Consider the problem for any fixed natural number m>2 of finding solutions to sum(k=1,n,k^m) = (k+1)^m. Erdos conjectured that there are no solutions for n,m>2. Let D be the matrix of differences of D[m,n] := sum(k=1,n,k^m) - (k+1)^m. Then the generating functions for the rows of this matrix D constitute a set of polynomials in n (for varying n along columns) and the m-th polynomial defining the m-th row. Let GF_D be the matrix of the coefficients of this set of polynomials. Then the present sequence is the (unsigned) second column of GF_D^-1. - Gottfried Helms, Apr 01 2007 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 J.-C. Aval, V. FĂ©ray, J.-C. Novelli, J.-Y. Thibon, Quasi-symmetric functions as polynomial functions on Young diagrams, arXiv preprint arXiv:1312.2727, 2013 Gottfried Helms, Discussion of a problem concerning summing of like powers FORMULA E.g.f.: (exp(x)-1)/(exp(x)*(2-exp(x))). O.g.f.: Sum_{n>=0} (2*n+1)! * x^(2*n+1) / Product_{k=1..2*n+1} (1-k*x). - Paul D. Hanna, Jul 20 2011 a(n)=Sum(Binomial(n, k)(-1)^(n-k)Sum(i! Stirling2(k, i), i=1, ..k), k=0, .., n). a(n) = (A000670(n)-(-1)^n)/2. - Vladeta Jovovic, Jan 17 2005 a(n) ~ n! / (4*(log(2))^(n+1)). - Vaclav Kotesovec, Feb 25 2014 a(n) = Sum_{k=0..floor(n/2)} (2*k+1)!*Stirling2(n, 2*k+1). - Peter Luschny, Sep 20 2015 MAPLE h := n -> add(combinat:-eulerian1(n, k)*2^k, k=0..n): a := n -> (h(n)-(-1)^n)/2: seq(a(n), n=0..20); # Peter Luschny, Jul 09 2015 MATHEMATICA Table[Sum[Binomial[n, k](-1)^(n-k)Sum[i! StirlingS2[k, i], {i, 1, k}], {k, 0, n}], {n, 0, 20}] PROG (PARI) a(n)=if(n<0, 0, n!*polcoeff(subst(y/(1-y^2), y, exp(x+x*O(x^n))-1), n)) (PARI) {a(n)=polcoeff(sum(m=0, n, (2*m+1)!*x^(2*m+1)/prod(k=1, 2*m+1, 1-k*x+x*O(x^n))), n)} /* Paul D. Hanna, Jul 20 2011 */ (Sage) def A089677_list(len):  # with a(0)=1     e, r = [1], [1]     for i in (1..len-1):         for k in range(i-1, -1, -1): e[k] = (e[k]*i)//(i-k)         r.append(-sum(e[j]*(-1)^(i-j) for j in (0..i-1)))         e.append(sum(e))     return r A089677_list(21) # Peter Luschny, Jul 09 2015 CROSSREFS Cf. A000670, A052841. Sequence in context: A100309 A198410 A199192 * A075996 A173766 A093168 Adjacent sequences:  A089674 A089675 A089676 * A089678 A089679 A089680 KEYWORD easy,nonn AUTHOR Mario Catalani (mario.catalani(AT)unito.it), Jan 03 2004 STATUS approved

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Last modified April 20 22:22 EDT 2019. Contains 322310 sequences. (Running on oeis4.)