%I A005425 M1461
%S A005425 1,2,5,14,43,142,499,1850,7193,29186,123109,538078,2430355,11317646,54229907,
%T A005425 266906858,1347262321,6965034370,36833528197,199037675054,1097912385851,
%U A005425 6176578272782,35409316648435,206703355298074,1227820993510153,7416522514174082
%N A005425 a(n) = 2*a(n-1)+(n-1)*a(n-2).
%C A005425 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 Penson, Apr 
               15 2004.
%C A005425 John W. Layman (layman(AT)math.vt.edu) observes that computationally 
               this agrees with the binomial transform of A000085.
%C A005425 Number of self-inverse partial permutations.
%C A005425 Number of '12-3 and 214-3'-avoiding permutations.
%C A005425 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 (deutsch(AT)duke.poly.edu), 
               Jan 10 2005
%C A005425 Comment from Franklin T. Adams-Watters, Dec 21 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<n} b(k)+k-n+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.
%C A005425 Number of n X n symmetric binary matrices with no row sum greater than 
               1. - Ron Hardin (rhhardin(AT)att.net), Jun 13 2008
%D A005425 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).
%D A005425 P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, 
               Combinatorial field theories via boson normal ordering, preprint, 
               Apr 27 2004.
%D A005425 R. Donaghey, Binomial self-inverse sequences and tangent coefficients, 
               J. Combin. Theory, Series A, 21 (1976), 155-163.
%D A005425 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A005425 T. D. Noe, <a href="b005425.txt">Table of n, a(n) for n=0..200</a>
%H A005425 P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, 
               <a href="http://arXiv.org/abs/quant-ph/0405103">Combinatorial field 
               theories via boson normal ordering</a>
%H A005425 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 
               4 (2001), #01.1.5.
%H A005425 T. Mansour, <a href="http://arXiv.org/abs/math.CO/0202219">Restricted 
               permutations by patterns of type 2-1</a>.
%F A005425 E.g.f.: exp (2 x + x^2 / 2 ).
%F A005425 a(n) = Sum_{k=0..n} binomial(n, k)*A000085(n-k) . - DELEHAM Philippe 
               (kolotoko(AT)wanadoo.fr), Mar 07 2004
%F A005425 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 (deutsch(AT)duke.poly.edu), 
               Nov 24 2004
%F A005425 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
%F A005425 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 
               (simoseve(AT)gmail.com), Apr 25 2007
%p A005425 with(orthopoly): seq((-I/sqrt(2))^n*H(n,I*sqrt(2)),n=0..25);
%t A005425 a[0] = 1; a[1] = 2; a[n_] := 2a[n - 1] + (n - 1)*a[n - 2]; Table[ a[n], 
               {n, 0, 25}] (* or *)
%t A005425 Range[0, 25]!CoefficientList[Series[Exp[2 x + x^2/2], {x, 0, 25}], x] 
               (* or *)
%t A005425 f[n_] := Sum[2^(n - 3k)n!/((n - 2k)!k!), {k, 0, n}]; Table[ f[n], {n, 
               0, 25}] (* or *)
%t A005425 Table[(-I/Sqrt[2])^n*HermiteH[n, I*Sqrt[2]], {n, 0, 25}] (from Robert 
               G. Wilson v (rgwv(at)rgwv.com), Nov 04 2005)
%Y A005425 a(n) = A027412(n+1)/2.
%Y A005425 Cf. A085483, A093620, A093620, A100862. Bisections give A093620, A100510.
%Y A005425 Cf. A111062.
%Y A005425 Sequence in context: A112808 A088927 A110489 this_sequence A035349 A155888 
               A006789
%Y A005425 Adjacent sequences: A005422 A005423 A005424 this_sequence A005426 A005427 
               A005428
%K A005425 nonn,easy,nice
%O A005425 0,2
%A A005425 Simon Plouffe (simon.plouffe(AT)gmail.com)
%E A005425 Recurrence and formula corrected Oct 15 1997 (Olivier Gerard).