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

1, 2, 5, 14, 43, 142, 499, 1850, 7193, 29186, 123109, 538078, 2430355, 11317646, 54229907, 266906858, 1347262321, 6965034370, 36833528197, 199037675054, 1097912385851, 6176578272782, 35409316648435, 206703355298074, 1227820993510153, 7416522514174082

Offset: 0

Author

Simon Plouffe

Keywords

• nonn
• easy
• nice

Comments

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

John W. Layman observes that computationally this agrees with the binomial transform of A000085.

Number of self-inverse partial permutations.

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

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

Consider finite sequences of positive integers of length n with b(1)=1 and with the constraint that b(m) <= max_{0A111062. - Franklin T. Adams-Watters, Dec 21 2005, corrected Dec 31 2014

Number of n X n symmetric binary matrices with no row sum greater than 1. - R. H. Hardin, Jun 13 2008

Polynomials in A099174 evaluated at x=2 (see also formula by Deutsch below). - Johannes W. Meijer, Feb 04 2010

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

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

From Robert A. Russell, Apr 28 2018: (Start)

Stirling transform of this sequence is A002872;

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

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

For a refined enumeration of the switchboard scenario presented by Penson above and in Donaghey and its relation to perfect matchings of simplices and an operator calculus, see A344678. - Tom Copeland, May 29 2021

Write [n] for {1, ..., n} and [n]^(k) for k-tuples without repeated entries. Then C [n]^(k) is naturally a complex S_n-representation, whose length is a(k) provided that n >= 2k. a(k) also gives the length of the countable dimensional Sym(N)-representation C N^(k), as remarked by Sam and Snowden (see link). - Jingjie Yang, Dec 28 2023

Examples

			From _Joerg Arndt_, Apr 18 2014: (Start)
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):
# :    word        partial involution
01:  [ . . . ]    ()
02:  [ . . 3 ]    (3)
03:  [ . 2 . ]    (2)
04:  [ . 2 2 ]    (2 3)
05:  [ . 2 3 ]    (2) (3)
06:  [ 1 . . ]    (1)
07:  [ 1 . 1 ]    (1 3)
08:  [ 1 . 3 ]    (1) (3)
09:  [ 1 1 . ]    (1 2)
10:  [ 1 1 3 ]    (1 2) (3)
11:  [ 1 2 . ]    (1) (2)
12:  [ 1 2 1 ]    (1 3) (2)
13:  [ 1 2 2 ]    (1) (2 3)
14:  [ 1 2 3 ]    (1) (2) (3)
(End)
		

References

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

Links

• T. D. Noe, Table of n, a(n) for n = 0..200
• E. Bagno and Y. Cherniavsky, Congruence B-orbits and the Bruhat poset of involutions of the symmetric group, Discrete Math., 312 (1) (2012), pp. 1289-1299.
• C. Banderier, M. Bousquet-Mélou, A. Denise, P. Flajolet, D. Gardy and D. Gouyou-Beauchamps, Generating Functions for Generating Trees, Discrete Mathematics 246(1-3), March 2002, pp. 29-55.
• A. Bingham and O. Ugurlu, Sects and lattice paths over the Lagrangian Grassmannian, arXiv preprint arXiv:1903.07229 [math.CO], 2019.
• P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, Combinatorial field theories via boson normal ordering, arXiv:quant-ph/0405103, 2004-2006. The title of this paper in the arXiv was later changed to "Some useful combinatorial formulas for bosonic operators"
• P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, Some useful combinatorial formulas for bosonic operators, J. Math. Phys. 46, 052110 (2005) (6 pages).
• R. Donaghey, Binomial self-inverse sequences and tangent coefficients, J. Combin. Theory, Series A, 21 (1976), 155-163.
• Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786 [math.CO], 2012. - From _N. J. A. Sloane_, Sep 17 2012
• T. Halverson and M. Reeks, Gelfand Models for Diagram Algebras, arXiv preprint arXiv:1302.6150 [math.RT], 2013-2014.
• J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
• Mikhail Khovanov, Victor Ostrik and Yakov Kononov, Two-dimensional topological theories, rational functions and their tensor envelopes, arXiv:2011.14758 [math.QA], 2020.
• T. Mansour, Restricted permutations by patterns of type 2-1, arXiv:math/0202219 [math.CO], 2002.
• D. Panyushev, On the orbits of a Borel subgroup in abelian ideals, arXiv preprint arXiv:1407.6857 [math.AG], 2014.
• Ville H. Pettersson, Enumerating Hamiltonian Cycles, The Electronic Journal of Combinatorics, Volume 21, Issue 4, 2014.
• K. Piejko, Extremal trees with respect to number of (A, B, 2C)-edge colourings, Journal of Applied Mathematics, Hindawi Publishing Corporation, Volume 2015, Article ID 463650, 5 pages.
• Steven V Sam and Andrew Snowden, Stability patterns in representation theory, arXiv:1302.5859 [math.RT], 2013-2015. See (1.3.4) p. 8.
• R. P. Stanley, On the enumeration of skew Young tableaux, Advances in Applied Mathematics 30 (2003) 283-294, (see corollary 2.4).

Crossrefs

Cf. A002872, A080337, A085483, A100862, A111062, A128227.

Bisections give A093620, A100510.

Row sums of A344678.

Programs

Haskell

a005425 n = a005425_list !! n
a005425_list = 1 : 2 : zipWith (+)
   (map (* 2) (tail a005425_list)) (zipWith (*) [1..] a005425_list)
-- Reinhard Zumkeller, Dec 18 2011

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

Maple

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

Mathematica

a[0] = 1; a[1] = 2; a[n_]:= 2a[n-1] + (n-1)*a[n-2]; Table[ a[n], {n, 0, 25}] (* or *)
Range[0, 25]!CoefficientList[Series[Exp[2 x + x^2/2], {x, 0, 25}], x] (* or *)
f[n_] := Sum[2^(n - 3k)n!/((n - 2k)!k!), {k, 0, n}]; Table[ f[n], {n, 0, 25}] (* or *)
Table[(-I/Sqrt[2])^n*HermiteH[n, I*Sqrt[2]], {n, 0, 25}] (* Robert G. Wilson v, Nov 04 2005 *)
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 *)
a[n_] := 2^(n/2) Exp[- I n Pi/2] HypergeometricU[-n/2, 1/2, -2];
Table[a[n], {n, 0, 22}] (* Peter Luschny, May 30 2021 *)

Maxima

makelist((%i/sqrt(2))^n*hermite(n,-%i*sqrt(2)),n,0,12); /* Emanuele Munarini, Mar 02 2016 */

PARI

A005425(n)=sum(k=0,n\2,2^(n-3*k)*n!/(n-2*k)!/k!) \\ M. F. Hasler, Jan 13 2012

SageMath

[(-i/sqrt(2))^n*hermite(n, i*sqrt(2)) for n in range(41)] # G. C. Greubel, Nov 19 2022

Formula

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

a(n) = A027412(n+1)/2. - N. J. A. Sloane, Sep 13 2003

a(n) = Sum_{k=0..n} binomial(n, k)*A000085(n-k). - Philippe Deléham, Mar 07 2004

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

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

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

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

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

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

a(n) ~ exp(2*sqrt(n) - n/2 - 1)*n^(n/2)/sqrt(2). - Vaclav Kotesovec, Oct 08 2012

a(n) = 2^(n/2)*exp(-i*n*Pi/2)*KummerU(-n/2, 1/2, -2). - Peter Luschny, May 30 2021

Extensions

Recurrence and formula corrected by Olivier Gérard, Oct 15 1997