A003724 Number of partitions of n-set into odd blocks.

1, 1, 1, 2, 5, 12, 37, 128, 457, 1872, 8169, 37600, 188685, 990784, 5497741, 32333824, 197920145, 1272660224, 8541537105, 59527313920, 432381471509, 3252626013184, 25340238127989, 204354574172160, 1699894200469849, 14594815769038848, 129076687233903673

Offset: 0

Author

R. H. Hardin

Keywords

• nonn
• nice
• easy

Examples

			G.f. = 1 + x + x^2 + 2*x^3 + 5*x^4 + 12*x^5 + 37*x^6 + 128*x^7 + 457*x^8 + ...
		

References

• L. Comtet, Analyse Combinatoire, Presses Univ. de France, 1970, Vol. II, pages 61-62.
• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 225, 2nd line of table.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..592 (first 101 terms from T. D. Noe)
• J. Riordan, Letter, Jul 06 1978
• Kruchinin Vladimir Victorovich, Composition of ordinary generating functions, arXiv:1009.2565 [math.CO], 2010.

Crossrefs

See A136630 for the table of partitions of an n-set into k odd blocks.

For partitions into even blocks see A005046 and A156289.

Cf. A000009, A000110. A002017, A009623.

Programs

Maple

a:= proc(n) option remember; `if`(n=0, 1, add(
      binomial(n-1, j-1)*irem(j, 2)*a(n-j), j=1..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 17 2015

Mathematica

a[n_] := Sum[((-1)^i*(k - 2*i)^n*Binomial[k, i])/(2^k*k!), {k, 1, n}, {i, 0, k}]; a[0] = 1; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Dec 21 2011, after Vladimir Kruchinin *)
With[{nn=30},CoefficientList[Series[Exp[Sinh[x]],{x,0,nn}],x]Range[0,nn]!] (* Harvey P. Dale, Apr 06 2012 *)
Table[Sum[BellY[n, k, Mod[Range[n], 2]], {k, 0, n}], {n, 0, 24}] (* Vladimir Reshetnikov, Nov 09 2016 *)

Maxima

a(n):=sum(1/2^k*sum((-1)^i*binomial(k,i)*(k-2*i)^n,i,0,k)/k!,k,1,n); /* Vladimir Kruchinin, Aug 22 2010 */

Formula

E.g.f.: exp ( sinh x ).

a(n) = sum(1/2^k*sum((-1)^i*C(k,i)*(k-2*i)^n, i=0..k)/k!, k=1..n). - Vladimir Kruchinin, Aug 22 2010

a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator sqrt(1+x^2)*d/dx. Cf. A002017 and A009623. - Peter Bala, Dec 06 2011

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} binomial(n-1,2*k) * a(n-2*k-1). - Ilya Gutkovskiy, Jul 11 2021

O.g.f A(X) satisfies A(x) = 1 + x*( A(x/(1-x))/(1-x) + A(x/(1+x))/(1+x) )/2. - Paul D. Hanna, Aug 19 2024