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
EXAMPLE
From Gus Wiseman, Jan 06 2021: (Start)
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)
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
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.
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.
KEYWORD
easy,nonn
AUTHOR
Mario Catalani (mario.catalani(AT)unito.it), Jan 03 2004
STATUS
approved