0,2

Values of Bell polynomials: ways of placing n labeled balls into n unlabeled (but 2-colored) boxes.

First column of the square of the matrix exp(P)/exp(1) given in A011971. - Gottfried Helms (helms(AT)uni-kassel.de), Mar 30 2007.

Base matrix in A011971, second power in A078937, third power in A078938, fourth power in A078939. - Gottfried Helms (helms(AT)uni-kassel.de), Apr 08 2007

T. S. Motzkin, Sorting numbers ...: for a link to this paper see A000262.

M. Aigner, A characterization of the Bell numbers, Discr. Math., 205 (1999), 207-210.

G. Labelle et al., Stirling numbers interpolation using permutations with forbidden subsequences, Discrete Math. 246 (2002), 177-195.

T. S. Motzkin, Sorting numbers for cylinders and other classification numbers, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176.

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

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 66

Zerinvary Lajos, Sage Notebooks

sum(2^k*stirling2(n, k), k=1..n). - Emeric Deutsch (deutsch(AT)duke.poly.edu), Oct 20 2001

a(n) = exp(-2)*sum(k>=1, 2^k*k^n/k! ) - Benoit Cloitre (benoit7848c(AT)orange.fr), Sep 25 2003

G.f. satisfies 2*(x/(1-x))*A(x/(1-x)) = A(x) - 1; twice the binomial transform equals the sequence shifted one place left. - Paul D. Hanna (pauldhanna(AT)juno.com), Dec 08 2003

With exact integer arithmetic (no infinite exp-sum needed): PE=exp(matpascal(5)-matid(6)); A = PE^2; a(n)=A[n,1]. - Gottfried Helms (helms(AT)uni-kassel.de), Apr 08 2007

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

sage: from sage.combinat.expnums import expnums2 sage: expnums(30, 2) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 26 2008

For boxes of 1 color, see A000110, for 3 colors see A027710.

First column of A078937. Equals 2*A035009(n), n>0.

Row sums of A033306, of A036073, and of A049020.

Cf. A000587, A002871, A068199, A068200, A068201.

Cf. A056857, A078937, A078938, A078944, A078945, A000110, A078937, A078938, A129323, A129324, A129325, A027710, A129327, A129328, A129329, A078944, A129331, A129332, A129333.

Sequence in context: A109317 A109153 A030453 this_sequence A049526 A093793 A087959

Adjacent sequences: A001858 A001859 A001860 this_sequence A001862 A001863 A001864

nonn,easy,nice

njas, Simon Plouffe (plouffe(AT)math.uqam.ca)