|
|
A003724
|
|
Number of partitions of n-set into odd blocks.
(Formerly M1427)
|
|
50
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
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
|
|
|
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:
|
|
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.
|
|
KEYWORD
|
nonn,nice,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|