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A003724 Number of partitions of n-set into odd blocks.
(Formerly M1427)
51
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
Alois P. Heinz, Table of n, a(n) for n = 0..592 (first 101 terms from T. D. Noe)
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.
Sequence in context: A355861 A002216 A024717 * A138314 A115277 A130221
KEYWORD
nonn,nice,easy
AUTHOR
STATUS
approved

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Last modified July 13 00:23 EDT 2024. Contains 374259 sequences. (Running on oeis4.)