cp's OEIS Frontend

A001861 Expansion of e.g.f. exp(2*(exp(x) - 1)).

%I A001861 M1662 N0653 #222 Apr 06 2024 09:03:11
%S A001861 1,2,6,22,94,454,2430,14214,89918,610182,4412798,33827974,273646526,
%T A001861 2326980998,20732504062,192982729350,1871953992254,18880288847750,
%U A001861 197601208474238,2142184050841734,24016181943732414,278028611833689478,3319156078802044158,40811417293301014150
%N A001861 Expansion of e.g.f. exp(2*(exp(x) - 1)).
%C A001861 Values of Bell polynomials: ways of placing n labeled balls into n unlabeled (but 2-colored) boxes.
%C A001861 First column of the square of the matrix exp(P)/exp(1) given in A011971. - _Gottfried Helms_, Mar 30 2007
%C A001861 Base matrix in A011971, second power in A078937, third power in A078938, fourth power in A078939. - _Gottfried Helms_, Apr 08 2007
%C A001861 Equals row sums of triangle A144061. - _Gary W. Adamson_, Sep 09 2008
%C A001861 Equals eigensequence of triangle A109128. - _Gary W. Adamson_, Apr 17 2009
%C A001861 Hankel transform is A108400. - _Paul Barry_, Apr 29 2009
%C A001861 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 A001861 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 A001861 Moments of the Poisson distribution with mean 2. - _Vladimir Reshetnikov_, May 17 2016
%C A001861 Exponential self-convolution of Bell numbers (A000110). - _Vladimir Reshetnikov_, Oct 06 2016
%D A001861 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A001861 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A001861 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 A001861 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 A001861 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 A001861 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 A001861 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 A001861 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 A001861 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 A001861 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 A001861 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 A001861 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 A001861 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=66">Encyclopedia of Combinatorial Structures 66</a> [broken link?]
%H A001861 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 A001861 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 A001861 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 A001861 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 A001861 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 A001861 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 A001861 OEIS Wiki, <a href="http://oeis.org/wiki/Sorting_numbers">Sorting numbers</a>
%H A001861 J. Riordan, <a href="/A001861/a001861.pdf">Letter to N. J. A. Sloane, Oct. 1970</a>
%H A001861 J. Riordan, <a href="/A001861/a001861_1.pdf">Letter, Oct 31 1977</a>
%H A001861 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 A001861 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 A001861 Jacob Sprittulla, <a href="https://arxiv.org/abs/2008.09984">On Colored Factorizations</a>, arXiv:2008.09984 [math.CO], 2020.
%F A001861 a(n) = Sum_{k=0..n} 2^k*Stirling2(n, k). - _Emeric Deutsch_, Oct 20 2001
%F A001861 a(n) = exp(-2)*Sum_{k>=1} 2^k*k^n/k!. - _Benoit Cloitre_, Sep 25 2003
%F A001861 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 A001861 PE = exp(matpascal(5)-matid(6)); A = PE^2; a(n)=A[n,1]. - _Gottfried Helms_, Apr 08 2007
%F A001861 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 A001861 O.g.f.: Sum_{n>=0} 2^n*x^n / Product_{k=1..n} (1-k*x). - _Paul D. Hanna_, Feb 15 2012
%F A001861 a(n) ~ exp(-2-n+n/LambertW(n/2))*n^n/LambertW(n/2)^(n+1/2). - _Vaclav Kotesovec_, Jan 06 2013
%F A001861 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 A001861 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 A001861 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 A001861 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 A001861 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 A001861 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 A001861 a(2) = 6: The six ways of putting 2 balls into bags (denoted by { }) and then into 2 labeled boxes (denoted by [ ]) are
%e A001861 01: [{1,2}] [ ];
%e A001861 02: [ ] [{1,2}];
%e A001861 03: [{1}] [{2}];
%e A001861 04: [{2}] [{1}];
%e A001861 05: [{1} {2}] [ ];
%e A001861 06: [ ] [{1} {2}].
%e A001861 - _Peter Bala_, Mar 23 2013
%p A001861 A001861:=n->add(Stirling2(n,k)*2^k, k=0..n); seq(A001861(n), n=0..20); # _Wesley Ivan Hurt_, Apr 18 2014
%p A001861 # second Maple program:
%p A001861 b:= proc(n, m) option remember;
%p A001861      `if`(n=0, 2^m, m*b(n-1, m)+b(n-1, m+1))
%p A001861     end:
%p A001861 a:= n-> b(n, 0):
%p A001861 seq(a(n), n=0..25);  # _Alois P. Heinz_, Aug 04 2021
%t A001861 Table[Sum[StirlingS2[n, k]*2^k, {k, 0, n}], {n, 0, 21}] (* _Geoffrey Critzer_, Oct 06 2009 *)
%t A001861 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 A001861 Table[BellB[n, 2], {n, 0, 20}] (* _Vaclav Kotesovec_, Jan 06 2013 *)
%o A001861 (PARI) a(n)=if(n<0,0,n!*polcoeff(exp(2*(exp(x+x*O(x^n))-1)),n))
%o A001861 (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 A001861 (PARI) {a(n) = sum(k=0, n, 2^k*stirling(n, k, 2))} \\ _Seiichi Manyama_, Jul 28 2019
%o A001861 (Sage) expnums(30, 2) # _Zerinvary Lajos_, Jun 26 2008
%o A001861 (Magma) [&+[2^k*StirlingSecond(n, k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, May 18 2019
%Y A001861 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 A001861 First column of A078937.
%Y A001861 Equals 2*A035009(n), n>0.
%Y A001861 Row sums of A033306, A036073, A049020, and A144061.
%Y A001861 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 A001861 nonn,easy,nice
%O A001861 0,2
%A A001861 _N. J. A. Sloane_, _Simon Plouffe_