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%I #50 Feb 11 2021 19:18:29
%S 0,1,1,7,37,271,2341,23647,272917,3543631,51123781,811316287,
%T 14045783797,263429174191,5320671485221,115141595488927,
%U 2657827340990677,65185383514567951,1692767331628422661,46400793659664205567,1338843898122192101557
%N Exponential convolution of A000670(n), with A000670(0)=0, with the sequence of all ones alternating in sign.
%C 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
%C 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
%H Vincenzo Librandi, <a href="/A089677/b089677.txt">Table of n, a(n) for n = 0..200</a>
%H J.-C. Aval, V. Féray, J.-C. Novelli, J.-Y. Thibon, <a href="http://arxiv.org/abs/1312.2727">Quasi-symmetric functions as polynomial functions on Young diagrams</a>, arXiv preprint arXiv:1312.2727, 2013
%H Gottfried Helms, <a href="http://go.helms-net.de/math/divers/ZerosOfGpFunctions.htm">Discussion of a problem concerning summing of like powers</a>
%F E.g.f.: (exp(x)-1)/(exp(x)*(2-exp(x))).
%F 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
%F a(n)=Sum(Binomial(n, k)(-1)^(n-k)Sum(i! Stirling2(k, i), i=1, ..k), k=0, .., n).
%F a(n) = (A000670(n)-(-1)^n)/2. - _Vladeta Jovovic_, Jan 17 2005
%F a(n) ~ n! / (4*(log(2))^(n+1)). - _Vaclav Kotesovec_, Feb 25 2014
%F a(n) = Sum_{k=0..floor(n/2)} (2*k+1)!*Stirling2(n, 2*k+1). - _Peter Luschny_, Sep 20 2015
%e From _Gus Wiseman_, Jan 06 2021: (Start)
%e a(n) is the number of ordered set partitions of {1..n} into an odd number of blocks. The a(1) = 1 through a(3) = 7 ordered set partitions are:
%e {{1}} {{1,2}} {{1,2,3}}
%e {{1},{2},{3}}
%e {{1},{3},{2}}
%e {{2},{1},{3}}
%e {{2},{3},{1}}
%e {{3},{1},{2}}
%e {{3},{2},{1}}
%e (End)
%p h := n -> add(combinat:-eulerian1(n,k)*2^k,k=0..n):
%p a := n -> (h(n)-(-1)^n)/2: seq(a(n),n=0..20); # _Peter Luschny_, Jul 09 2015
%t Table[Sum[Binomial[n, k](-1)^(n-k)Sum[i! StirlingS2[k, i], {i, 1, k}], {k, 0, n}], {n, 0, 20}]
%o (PARI) a(n)=if(n<0,0,n!*polcoeff(subst(y/(1-y^2),y,exp(x+x*O(x^n))-1),n))
%o (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 */
%o (Sage)
%o def A089677_list(len): # with a(0)=1
%o e, r = [1], [1]
%o for i in (1..len-1):
%o for k in range(i-1, -1, -1): e[k] = (e[k]*i)//(i-k)
%o r.append(-sum(e[j]*(-1)^(i-j) for j in (0..i-1)))
%o e.append(sum(e))
%o return r
%o A089677_list(21) # _Peter Luschny_, Jul 09 2015
%Y Ordered set partitions are counted by A000670.
%Y The case of (unordered) set partitions is A024429.
%Y The complement (even-length ordered set partitions) is counted by A052841.
%Y A058695 counts partitions of odd numbers, ranked by A300063.
%Y A101707 counts partitions of odd positive rank.
%Y A160786 counts odd-length partitions of odd numbers, ranked by A300272.
%Y A340102 counts odd-length factorizations into odd factors.
%Y A340692 counts partitions of odd rank.
%Y Other cases of odd length:
%Y - A027193 counts partitions of odd length.
%Y - A067659 counts strict partitions of odd length.
%Y - A166444 counts compositions of odd length.
%Y - A174726 counts ordered factorizations of odd length.
%Y - A332304 counts strict compositions of odd length.
%Y - A339890 counts factorizations of odd length.
%Y Cf. A000700, A026424, A027187, A028260, A078408, A174725, A236914, A340101.
%K easy,nonn
%O 0,4
%A Mario Catalani (mario.catalani(AT)unito.it), Jan 03 2004
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