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.

0, 1, 9, 39, 116, 275, 561, 1029, 1744, 2781, 4225, 6171, 8724, 11999, 16121, 21225, 27456, 34969, 43929, 54511, 66900, 81291, 97889, 116909, 138576, 163125, 190801, 221859, 256564, 295191, 338025, 385361, 437504, 494769, 557481, 625975, 700596

Offset

0,3

Comments

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

Starting from 1 the partial sums give A244864. - J. M. Bergot, Sep 17 2016

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Formula

a(n) = coefficient of x^4 in 1/((1-x)^n * (1-x^2)^binomial(n,2)).

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]

Example

From Michael B. Porter, Sep 18 2016: (Start)
The nine 2 X 2 matrices summing to 4 are:
4 0  3 0  2 0  1 0  0 0  2 1  1 1  0 1  0 2
0 0  0 1  0 2  0 3  0 4  1 0  1 1  1 2  2 0
(End)

Maple

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);

Mathematica

gf[k_] := 1/((1-x)^k (1-x^2)^(k(k-1)/2));
T[n_, k_] := SeriesCoefficient[gf[k], {x, 0, n}];
a[k_] := T[4, k];
a /@ Range[0, 40] (* Jean-François Alcover, Nov 07 2020 *)

Crossrefs

For 3 in place of 4 this gives A005900.

Row n=4 of A210391. - Alois P. Heinz, Mar 22 2012

Partial sums of A063489.

Sequence in context: A299280 A023163 A054121 * A034263 A374951 A060929

Adjacent sequences: A139591 A139592 A139593 * A139595 A139596 A139597

Keyword

easy,nonn

Author

Marc A. A. van Leeuwen, Jun 12 2008

Status

approved