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%I M1662 N0653 #222 Apr 06 2024 09:03:11
%S 1,2,6,22,94,454,2430,14214,89918,610182,4412798,33827974,273646526,
%T 2326980998,20732504062,192982729350,1871953992254,18880288847750,
%U 197601208474238,2142184050841734,24016181943732414,278028611833689478,3319156078802044158,40811417293301014150
%N Expansion of e.g.f. exp(2*(exp(x) - 1)).
%C Values of Bell polynomials: ways of placing n labeled balls into n unlabeled (but 2-colored) boxes.
%C First column of the square of the matrix exp(P)/exp(1) given in A011971. - _Gottfried Helms_, Mar 30 2007
%C Base matrix in A011971, second power in A078937, third power in A078938, fourth power in A078939. - _Gottfried Helms_, Apr 08 2007
%C Equals row sums of triangle A144061. - _Gary W. Adamson_, Sep 09 2008
%C Equals eigensequence of triangle A109128. - _Gary W. Adamson_, Apr 17 2009
%C Hankel transform is A108400. - _Paul Barry_, Apr 29 2009
%C The number of ways of putting n labeled balls into a set of bags and then putting the bags into 2 labeled boxes. An example is given below. - _Peter Bala_, Mar 23 2013
%C The f-vectors of n-dimensional hypercube are given by A038207 = exp[M*B(.,2)] = exp[M*A001861(.)] where M = A238385-I and (B(.,x))^n = B(n,x) are the Bell polynomials (cf. A008277). - _Tom Copeland_, Apr 17 2014
%C Moments of the Poisson distribution with mean 2. - _Vladimir Reshetnikov_, May 17 2016
%C Exponential self-convolution of Bell numbers (A000110). - _Vladimir Reshetnikov_, Oct 06 2016
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A001861/b001861.txt">Table of n, a(n) for n = 0..558</a> (terms 0..100 from T. D. Noe)
%H M. Aigner, <a href="http://dx.doi.org/10.1016/S0012-365X(99)00108-9">A characterization of the Bell numbers</a>, Discr. Math., 205 (1999), 207-210.
%H Michael Anshelevich, <a href="https://arxiv.org/abs/1708.08034">Product formulas on posets, Wick products, and a correction for the q-Poisson process</a>, arXiv:1708.08034 [math.OA], 2017, See Proposition 34 p. 25.
%H Diego Arcis, Camilo González, and Sebastián Márquez, <a href="https://arxiv.org/abs/2312.00574">Symmetric functions in noncommuting variables in superspace</a>, arXiv:2312.00574 [math.CO], 2023.
%H C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, <a href="http://dx.doi.org/10.1016/S0012-365X(01)00250-3">Generating Functions for Generating Trees</a>, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
%H J. M. Borwein, <a href="https://carmamaths.org/resources/jon/OEIStalk.pdf">Adventures with the OEIS: Five sequences Tony may like</a>, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016.
%H J. M. Borwein, <a href="/A060997/a060997.pdf">Adventures with the OEIS: Five sequences Tony may like</a>, Guttmann 70th [Birthday] Meeting, 2015, revised May 2016. [Cached copy, with permission]
%H Jacques Carlier and Corinne Lucet, <a href="http://dx.doi.org/10.1016/0166-218X(95)00032-M">A decomposition algorithm for network reliability evaluation</a>. In First International Colloquium on Graphs and Optimization (GOI), 1992 (Grimentz). Discrete Appl. Math. 65 (1996), 141-156 (see page 152 and Fig 6).
%H Adam M. Goyt and Lara K. Pudwell, <a href="http://arxiv.org/abs/1203.3786">Avoiding colored partitions of two elements in the pattern sense</a>, arXiv preprint arXiv:1203.3786 [math.CO], 2012. - From _N. J. A. Sloane_, Sep 17 2012
%H Wan-Ming Guo and Lily Li Liu, <a href="https://doi.org/10.2298/FIL2309923G">Asymptotic normality of the Stirling-Whitney-Riordan triangle</a>, Filomat (2023) Vol. 37, No. 9, 2923-2934.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=66">Encyclopedia of Combinatorial Structures 66</a> [broken link?]
%H Marin Knežević, Vedran Krčadinac, and Lucija Relić, <a href="https://arxiv.org/abs/2012.15307">Matrix products of binomial coefficients and unsigned Stirling numbers</a>, arXiv:2012.15307 [math.CO], 2020.
%H G. Labelle et al., <a href="http://dx.doi.org/10.1016/S0012-365X(01)00257-6">Stirling numbers interpolation using permutations with forbidden subsequences</a>, Discrete Math. 246 (2002), 177-195.
%H Huyile Liang, Jeffrey Remmel, and Sainan Zheng, <a href="https://arxiv.org/abs/1710.05795">Stieltjes moment sequences of polynomials</a>, arXiv:1710.05795 [math.CO], 2017, see page 20.
%H T. Mansour, M. Shattuck and D. G. L. Wang, <a href="http://arxiv.org/abs/1306.3355">Recurrence relations for patterns of type (2, 1) in flattened permutations</a>, arXiv preprint arXiv:1306.3355 [math.CO], 2013.
%H Victor Meally, <a href="/A002868/a002868.pdf">Comparison of several sequences given in Motzkin's paper "Sorting numbers for cylinders...", letter to N. J. A. Sloane, N. D.</a>
%H T. S. Motzkin, <a href="/A000262/a000262.pdf">Sorting numbers for cylinders and other classification numbers</a>, in Combinatorics, Proc. Symp. Pure Math. 19, AMS, 1971, pp. 167-176. [Annotated, scanned copy]
%H OEIS Wiki, <a href="http://oeis.org/wiki/Sorting_numbers">Sorting numbers</a>
%H J. Riordan, <a href="/A001861/a001861.pdf">Letter to N. J. A. Sloane, Oct. 1970</a>
%H J. Riordan, <a href="/A001861/a001861_1.pdf">Letter, Oct 31 1977</a>
%H Frank Simon, <a href="https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa-101154">Algebraic Methods for Computing the Reliability of Networks</a>, Dissertation, Doctor Rerum Naturalium (Dr. rer. nat.), Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden, 2012. See Table 5.1. - From _N. J. A. Sloane_, Jan 04 2013
%H Amit Kumar Singh, Akash Kumar and Thambipillai Srikanthan, <a href="http://www.ece.nus.edu.sg/stfpage/eleak/pdf/akumar_todaes_2012.pdf">Accelerating Throughput-aware Run-time Mapping for Heterogeneous MPSoCs</a>, ACM Transactions on Design Automation of Electronic Systems, 2012. - From _N. J. A. Sloane_, Dec 24 2012
%H Jacob Sprittulla, <a href="https://arxiv.org/abs/2008.09984">On Colored Factorizations</a>, arXiv:2008.09984 [math.CO], 2020.
%F a(n) = Sum_{k=0..n} 2^k*Stirling2(n, k). - _Emeric Deutsch_, Oct 20 2001
%F a(n) = exp(-2)*Sum_{k>=1} 2^k*k^n/k!. - _Benoit Cloitre_, Sep 25 2003
%F 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_, Dec 08 2003
%F PE = exp(matpascal(5)-matid(6)); A = PE^2; a(n)=A[n,1]. - _Gottfried Helms_, Apr 08 2007
%F 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). - _Paul Barry_, Apr 29 2009
%F O.g.f.: Sum_{n>=0} 2^n*x^n / Product_{k=1..n} (1-k*x). - _Paul D. Hanna_, Feb 15 2012
%F a(n) ~ exp(-2-n+n/LambertW(n/2))*n^n/LambertW(n/2)^(n+1/2). - _Vaclav Kotesovec_, Jan 06 2013
%F G.f.: (G(0) - 1)/(x-1)/2 where G(k) = 1 - 2/(1-k*x)/(1-x/(x-1/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jan 16 2013
%F G.f.: 1/Q(0) where Q(k) = 1 + x*k - x - x/(1 - 2*x*(k+1)/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Mar 07 2013
%F G.f.: ((1+x)/Q(0)-1)/(2*x), where Q(k) = 1 - (k+1)*x - 2*(k+1)*x^2/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 03 2013
%F G.f.: T(0)/(1-2*x), where T(k) = 1 - 2*x^2*(k+1)/( 2*x^2*(k+1) - (1-2*x-x*k)*(1-3*x-x*k)/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Oct 24 2013
%F a(n) = Sum_{k=0..n} A033306(n,k) = Sum_{k=0..n} binomial(n,k)*Bell(k)*Bell(n-k), where Bell = A000110 (see Motzkin, p. 170). - _Danny Rorabaugh_, Oct 18 2015
%F a(0) = 1 and a(n) = 2 * Sum_{k=0..n-1} binomial(n-1,k)*a(k) for n > 0. - _Seiichi Manyama_, Sep 25 2017 [corrected by _Ilya Gutkovskiy_, Jul 12 2020]
%e a(2) = 6: The six ways of putting 2 balls into bags (denoted by { }) and then into 2 labeled boxes (denoted by [ ]) are
%e 01: [{1,2}] [ ];
%e 02: [ ] [{1,2}];
%e 03: [{1}] [{2}];
%e 04: [{2}] [{1}];
%e 05: [{1} {2}] [ ];
%e 06: [ ] [{1} {2}].
%e - _Peter Bala_, Mar 23 2013
%p A001861:=n->add(Stirling2(n,k)*2^k, k=0..n); seq(A001861(n), n=0..20); # _Wesley Ivan Hurt_, Apr 18 2014
%p # second Maple program:
%p b:= proc(n, m) option remember;
%p `if`(n=0, 2^m, m*b(n-1, m)+b(n-1, m+1))
%p end:
%p a:= n-> b(n, 0):
%p seq(a(n), n=0..25); # _Alois P. Heinz_, Aug 04 2021
%t Table[Sum[StirlingS2[n, k]*2^k, {k, 0, n}], {n, 0, 21}] (* _Geoffrey Critzer_, Oct 06 2009 *)
%t mx = 16; p = 1; Range[0, mx]! CoefficientList[ Series[ Exp[ (Exp[p*x] - p - 1)/p + Exp[x]], {x, 0, mx}], x] (* _Robert G. Wilson v_, Dec 12 2012 *)
%t Table[BellB[n, 2], {n, 0, 20}] (* _Vaclav Kotesovec_, Jan 06 2013 *)
%o (PARI) a(n)=if(n<0,0,n!*polcoeff(exp(2*(exp(x+x*O(x^n))-1)),n))
%o (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 */
%o (PARI) {a(n) = sum(k=0, n, 2^k*stirling(n, k, 2))} \\ _Seiichi Manyama_, Jul 28 2019
%o (Sage) expnums(30, 2) # _Zerinvary Lajos_, Jun 26 2008
%o (Magma) [&+[2^k*StirlingSecond(n, k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, May 18 2019
%Y For boxes of 1 color, see A000110, for 3 colors see A027710, for 4 colors see A078944, for 5 colors see A144180, for 6 colors see A144223, for 7 colors see A144263, for 8 colors see A221159.
%Y First column of A078937.
%Y Equals 2*A035009(n), n>0.
%Y Row sums of A033306, A036073, A049020, and A144061.
%Y Cf. A000110, A000587, A002871, A027710, A056857, A068199, A068200, A068201, A078937, A078938, A078944, A078945, A109128, A129323, A129324, A129325, A129327, A129328, A129329, A129331, A129332, A129333, A144180, A144223, A144263, A189233, A213170, A221159, A221176.
%K nonn,easy,nice,changed
%O 0,2
%A _N. J. A. Sloane_, _Simon Plouffe_
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