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A053493 Number of symmetric 4 X 4 matrices of nonnegative integers with every row and column adding to n. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified December 8 08:22 EST 2022. Contains 358693 sequences. (Running on oeis4.)