0,2

Number of Boolean sublattices of the Boolean lattice of subsets of {1..n}.

a(n)=p(n+1) where p(x) is the unique degree-n polynomial such that p(k)=A000110(n+1) for k=0,1,...,n. - Michael Somos, Oct 07 2003

With offset 1, number of permutations beginning with 12 and avoiding 21-3.

Rows sums of Bell's triangle (A011971). - Jorge Coveiro (jorgecoveiro(AT)yahoo.com), Dec 26 2004

Number of blocks in all set partitions of an (n+1)-set. Example: a(2)=10 because the set partitions of {1,2,3} are 1|2|3, 1|23, 12|3, 13|2 and 123, with a total of 10 blocks. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 13 2006

Number of partitions of n+3 with at least one singleton and with the smallest element in a singleton equal to 2. - Olivier GERARD (olivier.gerard(AT)gmail.com), Oct 29 2007

Comment from Philippe Leroux, Nov 17 2007. See page 29, Theorem 5.6 of my paper on the arXiv: These numbers are the dimensions of the homogeneous components of the operad called ComTrip associated with commutative triplicial algebras. (Triplicial algebras are related to even trees and also to L-algebras, see A006013.

Number of set partitions of (n+2) elements where two specific elements are clustered separately. Example: a(1)=3 because 1/2/3, 1/23, 13/2 are the 3 set partitions with 1, 2 clustered separately. - Andrey Goder (andy.goder(AT)gmail.com), Dec 17 2007

Equals A008277 * [1,2,3,...], i.e. the product of the Stirling Number of the second kind triangle and the natural number vector. a(n+1) = row sums of triangle A137650 - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 31 2008

Prefaced with a "1" = row sums of triangle A152433. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 04 2008]

Equals row sums of triangle A159573 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 16 2009]

Olivier Gerard and Karol A. Penson, A budget of set partition statistics, in preparation.

S. Getu et al., How to guess a generating function, SIAM J. Discrete Math., 5 (1992), 497-499.

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

E. G. Whitehead, Jr., Stirling number identities from chromatic polynomials, J. Combin. Theory, A 24 (1978), 314-317.

T. D. Noe, Table of n, a(n) for n=0..100

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 152

S. Kitaev, Generalized pattern avoidance with additional restrictions, Sem. Lothar. Combinat. B48e (2003).

S. Kitaev and T. Mansour, Simultaneous avoidance of generalized patterns.

J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Philippe Leroux, An equivalence of categories motivated by weighted directed graphs

A. M. Odlyzko, Asymptotic enumeration methods, pp. 1063-1229 of R. L. Graham et al., eds., Handbook of Combinatorics, 1995; see Example 12.16 (pdf, ps)

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

Eric Weisstein's World of Mathematics, Bell Triangle

a(n-1) = Sum_{k=1..n} k*Stirling2(n, k) for n>=1.

E.g.f.: exp(exp(x) + 2*x - 1). First differences of Bell numbers (if offset 1). - Michael Somos, Oct 09, 2002.

G.f.: sum{k>=0, x^k/prod[l=1..k, 1-(l+1)x]}. - R. Stephan, Apr 18 2004

a(n) = Sum_{i=0..n} 2^(n-i)*B(i)*binomial(n,i) where B(n) = Bell numbers A000110(n). - Fred Lunnon, Aug 04 2007. Written umbrally, a(n) = (B+2)^n. - N. J. A. Sloane (njas(AT)research.att.com), Feb 07 2009.

Representation as an infinite series: a(n-1)=sum(k^n*(k-1)/k!, k=2..infinity)/exp(1), n=1, 2... This is a Dobinski-type summation formula. - Karol A. Penson (penson(AT)lptl.jussieu.fr), Mar 14, 2002.

Row sums of A011971 (Aitken's array, also called Bell triangle) - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Nov 15 2003

a(n) = EXP(-1)*sum(k=>0, (k+2)^(n)/k!) - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), Jun 03 2004

Recurrence : a(n+1) = 1 + sum { j=1, n, (1+binomial(n, j))*a(j) } - Jon Perry (perry(AT)globalnet.co.uk), Apr 25 2005

a(n) = A000296(n+3) - A000296(n+1) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 31 2005

a(n) = B(n+2) - B(n+1), where B(n) are Bell numbers (A000110). - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jul 13 2006

a(n) = A123158(n,2) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 06 2006

Binomial transform of Bell numbers 1, 2, 5, 15, 52, 203, 877, 4140,... (see A000110).

Define f_1(x),f_2(x),... such that f_1(x)=x*e^x, f_{n+1}(x)=diff(x*f_n(x),x), for n=2,3,.... Then a(n-1)=e^{-1}*f_n(1). - Milan R. Janjic (agnus(AT)blic.net), May 30 2008

Representation of numbers a(n), n=0,1..., as special values of hypergeometric function of type (n)F(n), in Maple notation: a(n)=exp(-1)*2^n*hypergeom([3,3...3],[2.2...2],1), n=0,1..., i.e. having n parameters all equal to 3 in the numerator, having n parameters all equal to 2 in the denominator and the value of the argument equal to 1. Examples: a(0)= 2^0*evalf(hypergeom([],[],1)/exp(1))=1 a(1)= 2^1*evalf(hypergeom([3],[2],1)/exp(1))=3 a(2)= 2^2*evalf(hypergeom([3,3],[2,2],1)/exp(1))=10 a(3)= 2^3*evalf(hypergeom([3,3,3],[2,2,2],1)/exp(1))=37 a(4)= 2^4*evalf(hypergeom([3,3,3,3],[2,2,2,2],1)/exp(1))=151 a(5)= 2^5*evalf(hypergeom([3,3,3,3,3],[2,2,2,2,2],1)/exp(1))= 674 - Karol A. Penson (penson(AT)lptl.jussieu.fr), Sep 28 2007

with(combinat): seq(bell(n+2)-bell(n+1), n=0..22); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 13 2006

seq(add(binomial(n, k)*(bell(n-k)), k=1..n), n=1..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Dec 01 2006

(PARI) a(n)=if(n<0, 0, n!*polcoeff(exp(exp(x+x*O(x^n))+2*x-1), n))

(PARI) a(n)=if(n<0, 0, n+=2; subst(polinterpolate(Vec(serlaplace(exp(exp(x+O(x^n))-1)-1))), x, n))

Cf. A000110, A005494, A049020, A011968, A011971.

Cf. A008277, A137650.

A152433 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 04 2008]

A159573 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 16 2009]

Sequence in context: A086444 A064613 A138378 this_sequence A123636 A092816 A078109

Adjacent sequences: A005490 A005491 A005492 this_sequence A005494 A005495 A005496

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)

Definition revised by David Callan (callan(AT)stat.wisc.edu), Oct 11 2005