A053493 Number of symmetric 4 X 4 matrices of nonnegative integers with every row and column adding to n.

1, 10, 56, 214, 641, 1620, 3616, 7340, 13825, 24510, 41336, 66850, 104321, 157864, 232576, 334680, 471681, 652530, 887800, 1189870, 1573121, 2054140, 2651936, 3388164, 4287361, 5377190, 6688696, 8256570, 10119425, 12320080, 14905856, 17928880, 21446401

Offset: 0

Author

N. J. A. Sloane, Jan 15 2000; definition revised Jul 06 2014

Keywords

• nonn
• easy

References

• R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986; see Prop. 4.6.21, p. 235, G_4(lambda).

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• L. Carlitz, Enumeration of symmetric arrays, Duke Math. J., Vol. 33 (1966), 771-782. MR0201332 (34 #1216).
• R. P. Stanley, Magic labelings of graphs, symmetric magic squares,..., Duke Math. J. 43 (3) (1976) 511-531, F_4(x) in section 5.
• Index entries for linear recurrences with constant coefficients, signature (6,-14,14,0,-14,14,-6,1).

Crossrefs

Cf. A019298, A053494.

Programs

Mathematica

CoefficientList[Series[(1+4x+10x^2+4x^3+x^4)/((1-x)^7(1+x)), {x, 0, 30}], x] (* or *) LinearRecurrence[{6,-14,14,0,-14,14,-6,1},{1,10,56,214,641,1620,3616,7340},30] (* Harvey P. Dale, Oct 31 2011 *)

PARI

Vec((1+4*x+10*x^2+4*x^3+x^4) / ((1-x)^7*(1+x)) + O(x^40)) \\ Colin Barker, Jan 14 2017

Formula

G.f.: (1+4*x+10*x^2+4*x^3+x^4)/((1-x)^7*(1+x)).

a(0)=1, a(1)=10, a(2)=56, a(3)=214, a(4)=641, a(5)=1620, a(6)=3616, a(7)=7340, a(n) = 6*a(n-1) - 14*a(n-2) + 14*a(n-3) - 14*a(n-5) + 14*a(n-6) - 6*a(n-7) + a(n-8). - Harvey P. Dale, Oct 31 2011

a(n) = (9*(31+(-1)^n) + 768*n + 928*n^2 + 624*n^3 + 238*n^4 + 48*n^5 + 4*n^6) / 288. - Colin Barker, Jan 14 2017