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A001861
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Expansion of exp {2 (exp(x) - 1)}.
(Formerly M1662 N0653)
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36
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1, 2, 6, 22, 94, 454, 2430, 14214, 89918, 610182, 4412798, 33827974, 273646526, 2326980998, 20732504062, 192982729350, 1871953992254, 18880288847750, 197601208474238, 2142184050841734, 24016181943732414, 278028611833689478
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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
Equals row sums of triangle A144061 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 09 2008]
Equals eigensequence of triangle A109128 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 17 2009]
Hankel transform is A108400. [From Paul Barry (pbarry(AT)wit.ie), Apr 29 2009]
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REFERENCES
| M. Aigner, A characterization of the Bell numbers, Discr. Math., 205 (1999), 207-210.
C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, [http://algo.inria.fr/banderier/Papers/DiscMath99.ps Generating Functions for Generating Trees], Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
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. S. Motzkin, Sorting numbers ...: for a link to this paper see A000262.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 66
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FORMULA
| 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
G.f.: 1/(1-2x-2x^2/(1-3x-4x^2/(1-4x-6x^2/(1-5x-8x^2/(1-6x-10x^2/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Apr 29 2009]
O.g.f.: Sum_{n>=0} 2^n*x^n / Product_{k=1..n} (1-k*x). [From Paul D. Hanna, Feb 15 2012]
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MATHEMATICA
| Table[Sum[StirlingS2[n, k]*2^k, {k, 0, n}], {n, 0, 21}] [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Oct 06 2009]
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PROG
| (PARI) a(n)=if(n<0, 0, n!*polcoeff(exp(2*(exp(x+x*O(x^n))-1)), n))
(PARI) {a(n)=polcoeff(sum(m=0, n, 2^m*x^m/prod(k=1, m, 1-k*x +x*O(x^n))), n)} /* Paul D. Hanna, Feb 15 2012 */
sage: from sage.combinat.expnums import expnums2 sage: expnums(30, 2) - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 26 2008
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CROSSREFS
| 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.
A144061 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Sep 09 2008]
A109128 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Apr 17 2009]
Sequence in context: A109317 A109153 A030453 * A049526 A187251 A193763
Adjacent sequences: A001858 A001859 A001860 * A001862 A001863 A001864
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KEYWORD
| nonn,easy,nice,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Simon Plouffe (simon.plouffe(AT)gmail.com)
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