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A005425 a(n) = 2*a(n-1) + (n-1)*a(n-2).
(Formerly M1461)
33
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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 <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
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
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
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.
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).
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
EXAMPLE
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)
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 *)
PROG
(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
(PARI) A005425(n)=sum(k=0, n\2, 2^(n-3*k)*n!/(n-2*k)!/k!) \\ M. F. Hasler, Jan 13 2012
(Maxima) makelist((%i/sqrt(2))^n*hermite(n, -%i*sqrt(2)), n, 0, 12); /* Emanuele Munarini, Mar 02 2016 */
(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
(SageMath) [(-i/sqrt(2))^n*hermite(n, i*sqrt(2)) for n in range(41)] # G. C. Greubel, Nov 19 2022
CROSSREFS
Bisections give A093620, A100510.
Row sums of A344678.
Sequence in context: A369158 A088927 A110489 * A035349 A155888 A366099
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
Recurrence and formula corrected by Olivier Gérard, Oct 15 1997
STATUS
approved

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Last modified April 20 05:55 EDT 2024. Contains 371799 sequences. (Running on oeis4.)