A003724 Number of partitions of n-set into odd blocks. (Formerly M1427)

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

0,4

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.

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

Example

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

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 *)

Prog

(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 */

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.

Sequence in context: A355861 A002216 A024717 * A138314 A115277 A130221

Adjacent sequences: A003721 A003722 A003723 * A003725 A003726 A003727

Keyword

nonn,nice,easy

Author

R. H. Hardin

Status

approved