A001495 Number of symmetric 0-1 (n+1) X (n+1) matrices with row sums 2 and first row starting 1,1 for n > 0, a(0)=1. (Formerly M2947 N1188)

1, 1, 1, 3, 13, 70, 462, 3592, 32056, 322626, 3611890, 44491654, 597714474, 8693651092, 136059119332, 2279212812480, 40681707637888, 770631412413148, 15438647456063004, 326091322648369684, 7241563996136849260, 168657537987709667976, 4110364564664358194536

Offset

0,4

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

H. Gupta, Enumeration of symmetric matrices, Duke Math. J., 35 (1968), vol 3, 653-659.

H. Gupta, Enumeration of symmetric matrices (annotated scanned copy)

Formula

It appears that e.g.f. = 1 + Integral_{t = 0..x} ((1-t)^(-3/2)*exp( t*(t^2+3*t-2)/(4-4*t) ). - Mark van Hoeij, Oct 25 2011

Recursion: a(n) = (n-1) a(n-1) + (n-2)^2 a(n-2) - (n-2)(n-3)(n-4) a(n-3) - (1/2) (n-2)(n-3)(n-4) a(n-4) - (1/2)(n-2)(n-3)(n-4)(n-5) a(n-5). - Robert Israel, Aug 05 2013

a(n) ~ exp(sqrt(2*n)-n-3/2) * n^(n-1/2) * (1+31/(24*sqrt(2*n))). - Vaclav Kotesovec, Aug 14 2013

Example

a(3) = 3 because there are 3 symmetric 4 X 4 0-1 matrices with row sums 2 and first row 1 1 0 0, namely
  1100,  1100,  1100,
  1001,  1010,  1100,
  0011,  0101,  0011,
  0110,  0011,  0011.

Maple

a:= proc(n) a(n):= `if`(n<2, 1, (n-1) *a(n-1) +(n-2)^2 *a(n-2) -
       (n-2)*(n-3)*(n-4)* a(n-3) - (1/2)* (n-2)*(n-3)*(n-4)* a(n-4) -
       (1/2)*(n-2)*(n-3)*(n-4)*(n-5)* a(n-5))
    end:
seq(a(n), n=0..30);  # Robert Israel, Aug 05 2013

Mathematica

max = 30; egf = 1 + Integrate[(1-t)^(-3/2)*Exp[t (t^2 + 3 t - 2)/(4 - 4 t)] + O[t]^max // Normal, t]; CoefficientList[egf, t]* Range[0, max]! (* Jean-François Alcover, Apr 06 2017, after Mark van Hoeij *)

Crossrefs

Sequence in context: A154677 A121586 A024337 * A284217 A213690 A227117

Adjacent sequences: A001492 A001493 A001494 * A001496 A001497 A001498

Keyword

nonn

Author

N. J. A. Sloane

Extensions

Better name from and edited by Robert Israel, Aug 05 2013

Status

approved