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%I A005493 M2851
%S A005493 1,3,10,37,151,674,3263,17007,94828,562595,3535027,23430840,163254885,
%T A005493 1192059223,9097183602,72384727657,599211936355,5150665398898,45891416030315,
%U A005493 423145657921379,4031845922290572,39645290116637023,401806863439720943
%N A005493 a(n) = number of partitions of [n+1] with a distinguished block. For 
               example, a(1) counts (12), (1)-2, 1-(2) where dashes separate blocks 
               and the distinguished block is parenthesized.
%C A005493 Number of Boolean sublattices of the Boolean lattice of subsets of {1..n}.
%C A005493 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
%C A005493 With offset 1, number of permutations beginning with 12 and avoiding 
               21-3.
%C A005493 Rows sums of Bell's triangle (A011971). - Jorge Coveiro (jorgecoveiro(AT)yahoo.com), 
               Dec 26 2004
%C A005493 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
%C A005493 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
%C A005493 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.
%C A005493 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
%C A005493 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
%C A005493 Prefaced with a "1" = row sums of triangle A152433. [From Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), Dec 04 2008]
%C A005493 Equals row sums of triangle A159573 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Apr 16 2009]
%D A005493 Olivier Gerard and Karol A. Penson, A budget of set partition statistics, 
               in preparation.
%D A005493 S. Getu et al., How to guess a generating function, SIAM J. Discrete 
               Math., 5 (1992), 497-499.
%D A005493 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005493 E. G. Whitehead, Jr., Stirling number identities from chromatic polynomials, 
               J. Combin. Theory, A 24 (1978), 314-317.
%H A005493 T. D. Noe, <a href="b005493.txt">Table of n, a(n) for n=0..100</a>
%H A005493 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=152">
               Encyclopedia of Combinatorial Structures 152</a>
%H A005493 S. Kitaev, <a href="http://www.mat.univie.ac.at/users/slc/public_html/
               wpapers/s48kitaev.html">Generalized pattern avoidance with additional 
               restrictions</a>, Sem. Lothar. Combinat. B48e (2003).
%H A005493 S. Kitaev and T. Mansour, <a href="http://arXiv.org/abs/math.CO/0205182">
               Simultaneous avoidance of generalized patterns</a>.
%H A005493 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 A005493 Philippe Leroux, <a href="http://xxx.lanl.gov/abs/0709.3453">An equivalence 
               of categories motivated by weighted directed graphs</a>
%H A005493 A. M. Odlyzko, Asymptotic enumeration methods, pp. 1063-1229 of R. L. 
               Graham et al., eds., Handbook of Combinatorics, 1995; see Example 
               12.16 (<a href="http://www.dtc.umn.edu/~odlyzko/doc/asymptotic.enum.pdf">
               pdf</a>, <a href="http://www.dtc.umn.edu/~odlyzko/doc/asymptotic.enum.ps">
               ps</a>)
%H A005493 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               StirlingTransform.html">Link to a section of The World of Mathematics.</
               a>
%H A005493 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               BellTriangle.html">Bell Triangle</a>
%F A005493 a(n-1) = Sum_{k=1..n} k*Stirling2(n, k) for n>=1.
%F A005493 E.g.f.: exp(exp(x) + 2*x - 1). First differences of Bell numbers (if 
               offset 1). - Michael Somos, Oct 09, 2002.
%F A005493 G.f.: sum{k>=0, x^k/prod[l=1..k, 1-(l+1)x]}. - R. Stephan, Apr 18 2004
%F A005493 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.
%F A005493 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.
%F A005493 Row sums of A011971 (Aitken's array, also called Bell triangle) - DELEHAM 
               Philippe (kolotoko(AT)wanadoo.fr), Nov 15 2003
%F A005493 a(n) = EXP(-1)*sum(k=>0, (k+2)^(n)/k!) - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), 
               Jun 03 2004
%F A005493 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
%F A005493 a(n) = A000296(n+3) - A000296(n+1) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Jul 31 2005
%F A005493 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
%F A005493 a(n) = A123158(n,2) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 
               06 2006
%F A005493 Binomial transform of Bell numbers 1, 2, 5, 15, 52, 203, 877, 4140,... 
               (see A000110).
%F A005493 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
%F A005493 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
%p A005493 with(combinat): seq(bell(n+2)-bell(n+1),n=0..22); - Emeric Deutsch (deutsch(AT)duke.poly.edu), 
               Nov 13 2006
%p A005493 seq(add(binomial(n, k)*(bell(n-k)), k=1..n), n=1..23); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Dec 01 2006
%o A005493 (PARI) a(n)=if(n<0,0,n!*polcoeff(exp(exp(x+x*O(x^n))+2*x-1),n))
%o A005493 (PARI) a(n)=if(n<0,0,n+=2; subst(polinterpolate(Vec(serlaplace(exp(exp(x+O(x^n))-1)-1))),
               x,n))
%Y A005493 Cf. A000110, A005494, A049020, A011968, A011971.
%Y A005493 Cf. A008277, A137650.
%Y A005493 A152433 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 04 2008]
%Y A005493 A159573 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 16 2009]
%Y A005493 Sequence in context: A086444 A064613 A138378 this_sequence A123636 A092816 
               A078109
%Y A005493 Adjacent sequences: A005490 A005491 A005492 this_sequence A005494 A005495 
               A005496
%K A005493 nonn,easy,nice
%O A005493 0,2
%A A005493 N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
%E A005493 Definition revised by David Callan (callan(AT)stat.wisc.edu), Oct 11 
               2005

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