A003724 - OEIS

%I M1427 #65 Aug 20 2024 09:14:04

%S 1,1,1,2,5,12,37,128,457,1872,8169,37600,188685,990784,5497741,

%T 32333824,197920145,1272660224,8541537105,59527313920,432381471509,

%U 3252626013184,25340238127989,204354574172160,1699894200469849,14594815769038848,129076687233903673

%N Number of partitions of n-set into odd blocks.

%D L. Comtet, Analyse Combinatoire, Presses Univ. de France, 1970, Vol. II, pages 61-62.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 225, 2nd line of table.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A003724/b003724.txt">Table of n, a(n) for n = 0..592</a> (first 101 terms from T. D. Noe)

%H J. Riordan, <a href="/A001850/a001850_2.pdf">Letter, Jul 06 1978</a>

%H Kruchinin Vladimir Victorovich, <a href="http://arxiv.org/abs/1009.2565">Composition of ordinary generating functions</a>, arXiv:1009.2565 [math.CO], 2010.

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

%F 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

%F 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

%F 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

%F 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

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

%p a:= proc(n) option remember; `if`(n=0, 1, add(

%p       binomial(n-1, j-1)*irem(j, 2)*a(n-j), j=1..n))

%p     end:

%p seq(a(n), n=0..30);  # _Alois P. Heinz_, Mar 17 2015

%t 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_ *)

%t With[{nn=30},CoefficientList[Series[Exp[Sinh[x]],{x,0,nn}],x]Range[0,nn]!] (* _Harvey P. Dale_, Apr 06 2012 *)

%t Table[Sum[BellY[n, k, Mod[Range[n], 2]], {k, 0, n}], {n, 0, 24}] (* _Vladimir Reshetnikov_, Nov 09 2016 *)

%o (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 */

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

%Y For partitions into even blocks see A005046 and A156289.

%Y Cf. A000009, A000110. A002017, A009623.

%K nonn,nice,easy

%O 0,4

%A _R. H. Hardin_