A006505 - OEIS

%I M4789 #71 Sep 22 2023 12:20:07

%S 1,0,0,1,1,1,11,36,92,491,2557,11353,60105,362506,2169246,13580815,

%T 91927435,650078097,4762023647,36508923530,292117087090,2424048335917,

%U 20847410586719,185754044235873,1711253808769653,16272637428430152

%N Number of partitions of an n-set into boxes of size >2.

%D J. Riordan, A budget of rhyme scheme counts, pp. 455 - 465 of Second International Conference on Combinatorial Mathematics, New York, April 4-7, 1978. Edited by Allan Gewirtz and Louis V. Quintas. Annals New York Academy of Sciences, 319, 1979.

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

%H Seiichi Manyama, <a href="/A006505/b006505.txt">Table of n, a(n) for n = 0..579</a> (terms 0..250 from Alois P. Heinz)

%H E. A. Enneking and J. C. Ahuja, <a href="http://www.fq.math.ca/Scanned/14-1/enneking.pdf">Generalized Bell numbers</a>, Fib. Quart., 14 (1976), 67-73.

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=102">Encyclopedia of Combinatorial Structures 102</a>

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

%H I. Mezo, <a href="http://arxiv.org/abs/1308.1637">Periodicity of the last digits of some combinatorial sequences</a>, arXiv preprint arXiv:1308.1637 [math.CO], 2013.

%H J. Riordan, <a href="/A005000/a005000.pdf">Cached copy of paper</a> [With permission]

%F E.g.f.: exp ( exp x - 1 - x - (1/2)*x^2 ).

%F a(n) = Sum_{k=1..[n/3]} A059022(n,k), n>=3. - _R. J. Mathar_, Nov 08 2008

%F a(n) = n! * sum(m=1..n, sum(k=0..m, k!*(-1)^(m-k) *binomial(m,k) *sum(i=0..n-m, stirling2(i+k,k) *binomial(m-k,n-m-i) *2^(-n+m+i)/ (i+k)!))/m!); a(0)=1. - _Vladimir Kruchinin_, Feb 01 2011

%F Define polynomials g_n by g_0=1, g_1=g_2=0, g_3=g_4=g_5=x; g(n) = x*Sum_{i=0..n-3} binomial(n-1,i)*g_i; then a(n) = g_n(1). [Riordan]

%F a(0) = 1; a(n) = Sum_{k=0..n-3} binomial(n-1,k+2) * a(n-k-3). - _Seiichi Manyama_, Sep 22 2023

%p Copy ZL := [ B,{B=Set(Set(Z, card>=3))}, labeled ]: [seq(combstruct[count](ZL, size=n), n=0..25)]; # _Zerinvary Lajos_, Mar 13 2007

%p G:={P=Set(Set(Atom,card>=3))}:combstruct[gfsolve](G,unlabeled,x):seq(combstruct[count]([P,G,labeled],size=i),i=0..25); # _Zerinvary Lajos_, Dec 16 2007

%p g:=proc(n) option remember; if n=0 then RETURN(1); fi; if n<=2 then RETURN(0); fi; if n<=5 then RETURN(x); fi; expand(x*add(binomial(n-1,i)*g(i),i=0..n-3)); end; [seq(subs(x=1,g(n)),n=0..60)]; # _N. J. A. Sloane_, Jul 20 2011

%t a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ Exp[ Exp @ x - 1 - x - x^2 / 2], {x, 0, n}]] (* _Michael Somos_, Jul 20 2011 *)

%t a[0] = 1; a[n_] := n!*Sum[Sum[k!*(-1)^(m-k)*Binomial[m, k]*Sum[StirlingS2[i+k, k]* Binomial[m-k, n-m-i]*2^(-n+m+i)/(i+k)!, {i, 0, n-m}], {k, 0, m}]/m!, {m, 1, n}]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Apr 03 2015, after _Vladimir Kruchinin_ *)

%t Table[Sum[(-1)^j * Binomial[n, j] * BellB[n-j] * 2^((j-1)/2) * HypergeometricU[(1 - j)/2, 3/2, 1/2], {j, 0, n}], {n, 0, 25}] (* _Vaclav Kotesovec_, Feb 09 2020 *)

%o (PARI) {a(n) = if( n<0, 0, n! * polcoeff( exp( exp( x + x * O(x^n)) - 1 - x - x^2 / 2), n))} /* _Michael Somos_, Jul 20 2011 */

%Y Column k=2 of A293024.

%Y Cf. A000110, A000296, A057814, A057837, A059022.

%Y Cf. A293038.

%K nonn,easy

%O 0,7

%A _N. J. A. Sloane_

%E More terms from _Christian G. Bower_, Nov 09 2000

%E Edited by _N. J. A. Sloane_, Jul 20 2011