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A089677
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Exponential convolution of A000670(n), with A000670(0)=0, with the sequence of all ones alternating in sign.
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10
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0, 1, 1, 7, 37, 271, 2341, 23647, 272917, 3543631, 51123781, 811316287, 14045783797, 263429174191, 5320671485221, 115141595488927, 2657827340990677, 65185383514567951, 1692767331628422661, 46400793659664205567, 1338843898122192101557
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OFFSET
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0,4
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COMMENTS
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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
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LINKS
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FORMULA
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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) = Sum_{k=0..floor(n/2)} (2*k+1)!*Stirling2(n, 2*k+1). - Peter Luschny, Sep 20 2015
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EXAMPLE
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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:
{{1}} {{1,2}} {{1,2,3}}
{{1},{2},{3}}
{{1},{3},{2}}
{{2},{1},{3}}
{{2},{3},{1}}
{{3},{1},{2}}
{{3},{2},{1}}
(End)
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MAPLE
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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
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MATHEMATICA
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Table[Sum[Binomial[n, k](-1)^(n-k)Sum[i! StirlingS2[k, i], {i, 1, k}], {k, 0, n}], {n, 0, 20}]
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PROG
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(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
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CROSSREFS
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Ordered set partitions are counted by A000670.
The case of (unordered) set partitions is A024429.
The complement (even-length ordered set partitions) is counted by A052841.
A101707 counts partitions of odd positive rank.
A160786 counts odd-length partitions of odd numbers, ranked by A300272.
A340102 counts odd-length factorizations into odd factors.
A340692 counts partitions of odd rank.
Other cases of odd length:
- A027193 counts partitions of odd length.
- A067659 counts strict partitions of odd length.
- A166444 counts compositions of odd length.
- A174726 counts ordered factorizations of odd length.
- A332304 counts strict compositions of odd length.
- A339890 counts factorizations of odd length.
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KEYWORD
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easy,nonn
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AUTHOR
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Mario Catalani (mario.catalani(AT)unito.it), Jan 03 2004
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STATUS
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approved
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