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 A005425 a(n) = 2*a(n-1) + (n-1)*a(n-2). (Formerly M1461) 29

%I M1461

%S 1,2,5,14,43,142,499,1850,7193,29186,123109,538078,2430355,11317646,

%T 54229907,266906858,1347262321,6965034370,36833528197,199037675054,

%U 1097912385851,6176578272782,35409316648435,206703355298074,1227820993510153,7416522514174082

%N a(n) = 2*a(n-1) + (n-1)*a(n-2).

%C Switchboard problem with n subscribers, where a subscriber who is not talking can be of either of two sexes. Subscribers who are talking cannot be distinguished by sex. See also A000085. - _Karol A. Penson_, Apr 15 2004

%C _John W. Layman_ observes that computationally this agrees with the binomial transform of A000085.

%C Number of self-inverse partial permutations.

%C Number of '12-3 and 214-3'-avoiding permutations.

%C Number of matchings of the corona K'(n) of the complete graph K(n) and the complete graph K(1); in other words, K'(n) is the graph constructed from K(n) by adding for each vertex v a new vertex v' and the edge vv'. Example: a(3)=14 because in the graph with vertex set {A,B,C,a,b,c} and edge set {AB,AC,BC,Aa,Bb,Cc} we have the following matchings: (i) the empty set (1); (ii) the edges as singletons (6); (iii) {Aa,BC},{Bb,AC},{Cc,AB},{Aa,Bb},{Aa,Cc}, {Bb,Cc} (6); (iv) {Aa,Bb,Cc} (1). Row sums of A100862. - _Emeric Deutsch_, Jan 10 2005

%C Consider finite sequences of positive integers <b(m)> of length n with b(1)=1 and with the constraint that b(m) <= max_{0<k<m} b(k)+k-m+2. The question is how many such sequences there are. (Note that when we consider only the term k=m-1, this becomes b(m) <= b(m-1)+1 and it is well known that the number of sequences under this constraint is the Catalan numbers.) This sequence starts (from n = 1): 1,2,5,14,43,142,499,1850,7193. This appears to be the present sequence. But I do not see any way to prove it. The number T(n,m) of sequences of length n which will limit the continuation to size n+1 to a maximum value of m+1 appears to be given by A111062. - _Franklin T. Adams-Watters_, Dec 21 2005, corrected Dec 31 2014

%C Number of n X n symmetric binary matrices with no row sum greater than 1. - _R. H. Hardin_, Jun 13 2008

%C Polynomials in A099174 evaluated at x=2 (see also formula by Deutsch below). - _Johannes W. Meijer_, Feb 04 2010

%C Equals eigensequence of triangle A128227. Example: a(5) = 142 = (1, 1, 2, 5, 14, 43) dot (1, 2, 3, 4, 5, 1) = (1 + 2 + 6 + 20 + 70 + 43); where (1, 2, 3, 4, 5, 1) = row 5 of triangle A128227. - _Gary W. Adamson_, Aug 27 2010

%C Number of words [d(1), d(2), ..., d(n)] where d(k) is either =0, or =k (a fixed point), or the only value repeating a previous fixed point, see example. - _Joerg Arndt_, Apr 18 2014

%C From _Robert A. Russell_, Apr 28 2018: (Start)

%C Stirling transform of this sequence is A002872;

%C Stirling transform of A005425(n-1) is A080337. (End)

%C Number of congruence orbits of upper-triangular n X n matrices on symmetric matrices, or the number of Borel orbits in largest sect of the type CI symmetric space Sp_{2n}(C)/GL_n(C). - _Aram Bingham_, Oct 10 2019

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A005425/b005425.txt">Table of n, a(n) for n = 0..200</a>

%H E. Bagno and Y. Cherniavsky, <a href="https://doi.org/10.1016/j.disc.2011.12.018">Congruence B-orbits and the Bruhat poset of involutions of the symmetric group</a>, Discrete Math., 312 (1) (2012), pp. 1289-1299.

%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 A. Bingham and O. Ugurlu, <a href="https://arxiv.org/abs/1903.07229">Sects and lattice paths over the Lagrangian Grassmannian</a>, arXiv preprint arXiv:1903.07229 [math.CO], 2019.

%H P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, <a href="https://arXiv.org/abs/quant-ph/0405103">Combinatorial field theories via boson normal ordering</a>, arXiv:quant-ph/0405103, 2004-2006.

%H P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, <a href="http://dx.doi.org/10.1063/1.1904161">Some useful combinatorial formulas for bosonic operators</a>, J. Math. Phys. 46, 052110 (2005) (6 pages).

%H R. Donaghey, <a href="http://dx.doi.org/10.1016/0097-3165(76)90060-1">Binomial self-inverse sequences and tangent coefficients</a>, J. Combin. Theory, Series A, 21 (1976), 155-163.

%H Adam M. Goyt and Lara K. Pudwell, <a href="https://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 T. Halverson and M. Reeks, <a href="https://arxiv.org/abs/1302.6150">Gelfand Models for Diagram Algebras</a>, arXiv preprint arXiv:1302.6150 [math.RT], 2013-2014.

%H J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.

%H Mikhail Khovanov, Victor Ostrik and Yakov Kononov, <a href="https://arxiv.org/abs/2011.14758">Two-dimensional topological theories, rational functions and their tensor envelopes</a>, arXiv:2011.14758 [math.QA], 2020.

%H T. Mansour, <a href="https://arxiv.org/abs/math/0202219">Restricted permutations by patterns of type 2-1</a>, arXiv:math/0202219 [math.CO], 2002.

%H D. Panyushev, <a href="https://arxiv.org/abs/1407.6857">On the orbits of a Borel subgroup in abelian ideals</a>, arXiv preprint arXiv:1407.6857 [math.AG], 2014.

%H Ville H. Pettersson, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v21i4p7">Enumerating Hamiltonian Cycles</a>, The Electronic Journal of Combinatorics, Volume 21, Issue 4, 2014.

%H K. Piejko, <a href="http://dx.doi.org/10.1155/2015/463650">Extremal trees with respect to number of (A, B, 2C)-edge colourings</a>, Journal of Applied Mathematics, Hindawi Publishing Corporation, Volume 2015, Article ID 463650, 5 pages.

%H R. P. Stanley, <a href="http://dx.doi.org/10.1016/S0196-8858(02)00537-7">On the enumeration of skew Young tableaux</a>, Advances in Applied Mathematics 30 (2003) 283-294, (see corollary 2.4).

%F E.g.f.: exp( 2*x + x^2/2 ).

%F a(n) = A027412(n+1)/2. - _N. J. A. Sloane_, Sep 13 2003

%F a(n) = Sum_{k=0..n} binomial(n, k)*A000085(n-k). - _Philippe Deléham_, Mar 07 2004

%F a(n) = (-i*sqrt(2))^n*H(n, i*sqrt(2)), where H(n, x) is the n-th Hermite polynomial and i = sqrt(-1). - _Emeric Deutsch_, Nov 24 2004

%F a(n) = Sum_{k=0..floor(n/2)} 2^{n-3*k}*n!/((n-2*k)!*k!). - Huajun Huang (hua_jun(AT)hotmail.com), Oct 10 2005

%F For all n, a(n) = [M_n]_1,1 = [M_n]_2,1, where M_n = A_n * A_n-1 * ... * A_1, being A_k the matrix A_k = [1, k;1, 1]. - _Simone Severini_, Apr 25 2007

%F a(n) = (1/sqrt(2*Pi))*Integral_{x=-infinity..infinity} exp(-x^2/2)*(x+2)^n. - _Groux Roland_, Mar 14 2011

%F G.f.: 1/(1-2x-x^2/(1-2x-2x^2/(1-2x-3x^2/(1-2x-4x^2/(1-... (continued fraction).

%F E.g.f.: G(0) where G(k) = 1 + x*(4+x)/(4*k + 2 - x*(4+x)*(4*k+2)/(x*(4+x) + 4*(k+1)/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Dec 17 2011

%F a(n) ~ exp(2*sqrt(n) - n/2 - 1)*n^(n/2)/sqrt(2). - _Vaclav Kotesovec_, Oct 08 2012

%e From _Joerg Arndt_, Apr 18 2014: (Start)

%e The a(3) = 14 words [d(1), d(2), d(3)] where d(k) is either =0, or =k (a fixed point), or the only value repeating a previous fixed point are (dots for zeros):

%e # : word partial involution

%e 01: [ . . . ] ()

%e 02: [ . . 3 ] (3)

%e 03: [ . 2 . ] (2)

%e 04: [ . 2 2 ] (2 3)

%e 05: [ . 2 3 ] (2) (3)

%e 06: [ 1 . . ] (1)

%e 07: [ 1 . 1 ] (1 3)

%e 08: [ 1 . 3 ] (1) (3)

%e 09: [ 1 1 . ] (1 2)

%e 10: [ 1 1 3 ] (1 2) (3)

%e 11: [ 1 2 . ] (1) (2)

%e 12: [ 1 2 1 ] (1 3) (2)

%e 13: [ 1 2 2 ] (1) (2 3)

%e 14: [ 1 2 3 ] (1) (2) (3)

%e (End)

%p with(orthopoly): seq((-I/sqrt(2))^n*H(n,I*sqrt(2)),n=0..25);

%t a[0] = 1; a[1] = 2; a[n_] := 2a[n - 1] + (n - 1)*a[n - 2]; Table[ a[n], {n, 0, 25}] (* or *)

%t Range[0, 25]!CoefficientList[Series[Exp[2 x + x^2/2], {x, 0, 25}], x] (* or *)

%t f[n_] := Sum[2^(n - 3k)n!/((n - 2k)!k!), {k, 0, n}]; Table[ f[n], {n, 0, 25}] (* or *)

%t Table[(-I/Sqrt[2])^n*HermiteH[n, I*Sqrt[2]], {n, 0, 25}] (* _Robert G. Wilson v_, Nov 04 2005 *)

%t RecurrenceTable[{a[0]==1,a[1]==2,a[n]==2a[n-1]+(n-1)a[n-2]},a,{n,30}] (* _Harvey P. Dale_, Sep 30 2015 *)

%o a005425 n = a005425_list !! n

%o a005425_list = 1 : 2 : zipWith (+)

%o (map (* 2) (tail a005425_list)) (zipWith (*) [1..] a005425_list)

%o -- _Reinhard Zumkeller_, Dec 18 2011

%o (PARI) A005425(n)=sum(k=0,n\2,2^(n-3*k)*n!/(n-2*k)!/k!) \\ _M. F. Hasler_, Jan 13 2012

%o (Maxima) makelist((%i/sqrt(2))^n*hermite(n,-%i*sqrt(2)),n,0,12); /* _Emanuele Munarini_, Mar 02 2016 */

%o (MAGMA) a:=[2,5];[1] cat [n le 2 select a[n] else 2*Self(n-1) + (n-1)*Self(n-2):n in [1..30]]; // _Marius A. Burtea_, Oct 10 2019

%Y Cf. A085483, A100862, A111062, A128227.

%Y Bisections give A093620, A100510.

%Y Cf. A002872, A080337.

%K nonn,easy,nice

%O 0,2

%A _Simon Plouffe_

%E Recurrence and formula corrected by _Olivier Gérard_, Oct 15 1997

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