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

0,2

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).

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

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 *)

Prog

(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

Crossrefs

Cf. A019298, A053494.

Sequence in context: A281207 A228888 A137931 * A198833 A268462 A296918

Adjacent sequences: A053490 A053491 A053492 * A053494 A053495 A053496

Keyword

nonn,easy

Author

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

Status

approved