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A001495
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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)
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1
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1, 1, 1, 3, 13, 70, 462, 3592, 32056, 322626, 3611890, 44491654, 597714474, 8693651092, 136059119332, 2279212812480, 40681707637888, 770631412413148, 15438647456063004, 326091322648369684, 7241563996136849260, 168657537987709667976, 4110364564664358194536
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OFFSET
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0,4
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REFERENCES
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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).
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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MAPLE
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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:
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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