%I A000296 M3423 N1387
%S A000296 1,0,1,1,4,11,41,162,715,3425,17722,98253,580317,3633280,24011157,
%T A000296 166888165,1216070380,9264071767,73600798037,608476008122,
%U A000296 5224266196935,46499892038437,428369924118314,4078345814329009
%N A000296 Number of partitions of an n-set into blocks of size >1. Also number 
               of cyclically spaced (or feasible) partitions.
%C A000296 a(n+2)=p(n+1) where p(x) is the unique degree-n polynomial such that 
               p(k)=A000110(n) for k=0,1,...,n. - Michael Somos, Oct 07 2003
%C A000296 Number of complete rhyming schemes.
%C A000296 Whereas the Bell number B(n) (A000110(n)) is the number of terms in the 
               polynomial that expresses the n-th moment of a probability distribution 
               as a function of the first n cumulants, these numbers give the number 
               of terms in the corresponding expansion of the _central_ moment as 
               a function of the first n cumulants. - Michael Hardy (hardy(AT)math.umn.edu), 
               Jan 26 2005
%C A000296 Row sums of the triangle of associated Stirling numbers of second kind 
               A008299 . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 10 2005
%C A000296 a(n) = number of permutations on [n] for which the left-to-right maxima 
               coincide with the descents (entries followed by a smaller number). 
               For example, a(4) counts 2143, 3142, 3241, 4123. - David Callan (callan(AT)stat.wisc.edu), 
               Jul 20 2005
%D A000296 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000296 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000296 E. Bach, Random bisection and evolutionary walks, J. Applied Probability, 
               v. 38, pp. 582-596, 2001.
%D A000296 H. D. Becker, Solution to problem E 461, American Math Monthly 48 (1941), 
               701-702.
%D A000296 F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 
               (1999) 73-112.
%D A000296 E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 
               14 (1976), 67-73.
%D A000296 Martin Gardner in Sci. Amer. May 1977.
%D A000296 G. P\'{o}lya and G. Szeg\"{o}, Problems and Theorems in Analysis, Springer-Verlag, 
               NY, 2 vols., 1972, Vol. 1, p. 228.
%D A000296 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.
%H A000296 T. D. Noe, <a href="b000296.txt">Table of n, a(n) for n=0..100</a>
%H A000296 S. R. Finch, <a href="http://algo.inria.fr/bsolve/">Moments of sums</
               a>
%H A000296 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=16">
               Encyclopedia of Combinatorial Structures 16</a>
%H A000296 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 
               4 (2001), #01.1.5.
%H A000296 J. Riordan, <a href="a005000.pdf">Cached copy of paper</a>
%H A000296 <a href="Sindx_Par.html#partN">Index entries for related partition-counting 
               sequences</a>
%F A000296 E.g.f.: exp(exp(x) - 1 - x).
%F A000296 B(n) = a(n) + a(n+1), where B = A000110 = Bell numbers [Becker]
%F A000296 Inverse binomial transform of Bell numbers (A000110).
%F A000296 a(n)= sum((k)^n/(k+1)!, k = -1 .. infinity)/exp(1). - Vladeta Jovovic 
               (vladeta(AT)eunet.rs) and Karol A. PENSON (penson(AT)lptl.jussieu.fr), 
               Feb 02 2003
%F A000296 a(n)= sum(((-1)^(n-k))*binomial(n, k)*Bell(k), k=0..n) = (-1)^n + Bell(n) 
               - A087650(n), with Bell(n)=A000110(n). Wolfdieter Lang (wolfdieter.lang_AT_physik_DOT_uni-karlsruhe_DOT_d\
               e), Dec 01 2003
%F A000296 O.g.f.: A(x) = 1/(1-0*x-1*x^2/(1-1*x-2*x^2/(1-2*x-3*x^2/(1-... -(n-1)*x-n*x^2/
               (1- ...))))) (continued fraction). - Paul D. Hanna (pauldhanna(AT)juno.com), 
               Jan 17 2006
%F A000296 a(n)= sum(k=0..n) {(-1)^(n-k)* sum(j=0..k)[(-1)^j * binomial(k,j)* (1-j)^n]/ 
               k!} = sum over row n of A105794 - Tom Copeland (tcjpn(AT)msn.com), 
               Jun 05 2006
%F A000296 a(n)=(-1)^n + sum[(-1)^(j-1)*B(n-j),j=1..n], where B(q) are the Bell 
               numbers (A000110). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 
               29 2006
%p A000296 spec := [ B, {B=Set(Set(Z,card>1))}, labeled ]; [seq(combstruct[count](spec, 
               size=n), n=0..30)];
%p A000296 with(combinat): a:=n->(-1)^n+sum((-1)^(j-1)*bell(n-j),j=1..n): seq(a(n),
               n=0..30); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 29 2006
%p A000296 f:=exp(exp(x)-1-x): fser:=series(f, x=0, 31): 1, seq(n!*coeff(fser, x^n), 
               n=1..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Nov 22 
               2006
%p A000296 G:={P=Set(Set(Atom,card>=2))}:combstruct[gfsolve](G,unlabeled,x):seq(combstruct[count]([P,
               G,labeled],size=i),i=0..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 
               Dec 16 2007
%o A000296 (PARI) a(n)=if(n<2,n==0,subst(polinterpolate(Vec(serlaplace(exp(exp(x+O(x^n)/
               x)-1)))),x,n))
%Y A000296 Cf. A000110, A006505, A057814, A057837.
%Y A000296 Cf. A105794.
%Y A000296 Sequence in context: A151455 A149269 A149270 this_sequence A032265 A151273 
               A149271
%Y A000296 Adjacent sequences: A000293 A000294 A000295 this_sequence A000297 A000298 
               A000299
%K A000296 nonn,easy
%O A000296 0,5
%A A000296 N. J. A. Sloane (njas(AT)research.att.com).
%E A000296 More terms, new description from Christian G. Bower (bowerc(AT)usa.net), 
               Nov 15 1999.
%E A000296 Becker reference from D. E. Knuth, Dec 20 2003