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A005425 a(n) = 2*a(n-1) + (n-1)*a(n-2).
(Formerly M1461)
26
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)

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

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.

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.

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.

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..[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))*int(exp(-x^2/2)*(x+2)^n,x=-infinity..infinity). - 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

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 *)

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 */

CROSSREFS

Cf. A085483, A093620, A093620, A100862, A111062, A128227.

Bisections give A093620, A100510.

Cf. A002872, A080337.

Sequence in context: A112808 A088927 A110489 * A035349 A155888 A254314

Adjacent sequences:  A005422 A005423 A005424 * A005426 A005427 A005428

KEYWORD

nonn,easy,nice

AUTHOR

Simon Plouffe

EXTENSIONS

Recurrence and formula corrected by Olivier Gérard, Oct 15 1997

STATUS

approved

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Last modified May 20 17:41 EDT 2018. Contains 304340 sequences. (Running on oeis4.)