0,2

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.

John W. Layman (layman(AT)math.vt.edu) 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 (deutsch(AT)duke.poly.edu), Jan 10 2005

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.

Number of n X n symmetric binary matrices with no row sum greater than 1. - Ron Hardin (rhhardin(AT)att.net), Jun 13 2008

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).

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.

R. Donaghey, Binomial self-inverse sequences and tangent coefficients, J. Combin. Theory, Series A, 21 (1976), 155-163.

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

T. D. Noe, Table of n, a(n) for n=0..200

P. Blasiak, K. A. Penson, A. I. Solomon, A. Horzela and G. E. H. Duchamp, Combinatorial field theories via boson normal ordering

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.

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

a(n) = Sum_{k=0..n} binomial(n, k)*A000085(n-k) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), 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 (deutsch(AT)duke.poly.edu), 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 (simoseve(AT)gmail.com), Apr 25 2007

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

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}] (from Robert G. Wilson v (rgwv(at)rgwv.com), Nov 04 2005)

a(n) = A027412(n+1)/2.

Cf. A085483, A093620, A093620, A100862. Bisections give A093620, A100510.

Cf. A111062.

Sequence in context: A112808 A088927 A110489 this_sequence A035349 A155888 A006789

Adjacent sequences: A005422 A005423 A005424 this_sequence A005426 A005427 A005428

nonn,easy,nice

Simon Plouffe (simon.plouffe(AT)gmail.com)

Recurrence and formula corrected Oct 15 1997 (Olivier Gerard).