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A139594 Number of different n X n symmetric matrices with nonnegative entries summing to 4. Also number of symmetric oriented graphs with 4 arcs on n points.

%I A139594 #30 Jul 18 2022 16:38:19
%S A139594 0,1,9,39,116,275,561,1029,1744,2781,4225,6171,8724,11999,16121,21225,
%T A139594 27456,34969,43929,54511,66900,81291,97889,116909,138576,163125,
%U A139594 190801,221859,256564,295191,338025,385361,437504,494769,557481,625975,700596
%N A139594 Number of different n X n symmetric matrices with nonnegative entries summing to 4. Also number of symmetric oriented graphs with 4 arcs on n points.
%C A139594 a(n) is also the number of semistandard Young tableaux over all partitions of 4 with maximal element <= n. - _Alois P. Heinz_, Mar 22 2012
%C A139594 Starting from 1 the partial sums give A244864. - _J. M. Bergot_, Sep 17 2016
%H A139594 Alois P. Heinz, <a href="/A139594/b139594.txt">Table of n, a(n) for n = 0..1000</a>
%F A139594 a(n) = coefficient of x^4 in 1/((1-x)^n * (1-x^2)^binomial(n,2)).
%F A139594 a(n) = (n^2*(7+5*n^2))/12. G.f.: x*(1+x)*(1+3*x+x^2)/(1-x)^5. [_Colin Barker_, Mar 18 2012]
%e A139594 From _Michael B. Porter_, Sep 18 2016: (Start)
%e A139594 The nine 2 X 2 matrices summing to 4 are:
%e A139594 4 0  3 0  2 0  1 0  0 0  2 1  1 1  0 1  0 2
%e A139594 0 0  0 1  0 2  0 3  0 4  1 0  1 1  1 2  2 0
%e A139594 (End)
%p A139594 dd := proc(n,m) coeftayl(1/((1-X)^m*(1-X^2)^binomial(m,2)),X=0,n); seq(dd(4,m),m=0..N);
%t A139594 gf[k_] := 1/((1-x)^k (1-x^2)^(k(k-1)/2));
%t A139594 T[n_, k_] := SeriesCoefficient[gf[k], {x, 0, n}];
%t A139594 a[k_] := T[4, k];
%t A139594 a /@ Range[0, 40] (* _Jean-François Alcover_, Nov 07 2020 *)
%Y A139594 For 3 in place of 4 this gives A005900.
%Y A139594 Row n=4 of A210391. - _Alois P. Heinz_, Mar 22 2012
%Y A139594 Partial sums of A063489.
%K A139594 easy,nonn
%O A139594 0,3
%A A139594 _Marc A. A. van Leeuwen_, Jun 12 2008