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%I #28 Jan 14 2017 13:18:32
%S 1,10,56,214,641,1620,3616,7340,13825,24510,41336,66850,104321,157864,
%T 232576,334680,471681,652530,887800,1189870,1573121,2054140,2651936,
%U 3388164,4287361,5377190,6688696,8256570,10119425,12320080,14905856,17928880,21446401
%N Number of symmetric 4 X 4 matrices of nonnegative integers with every row and column adding to n.
%D R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986; see Prop. 4.6.21, p. 235, G_4(lambda).
%H Colin Barker, <a href="/A053493/b053493.txt">Table of n, a(n) for n = 0..1000</a>
%H L. Carlitz, <a href="http://dx.doi.org/10.1215/S0012-7094-66-03392-8">Enumeration of symmetric arrays</a>, Duke Math. J., Vol. 33 (1966), 771-782. MR0201332 (34 #1216).
%H R. P. Stanley, <a href="https://pdfs.semanticscholar.org/6054/5d22b1c1ec269264a717b18f3e3d8b346f99.pdf">Magic labelings of graphs, symmetric magic squares,...</a>, Duke Math. J. 43 (3) (1976) 511-531, F_4(x) in section 5.
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (6,-14,14,0,-14,14,-6,1).
%F G.f.: (1+4*x+10*x^2+4*x^3+x^4)/((1-x)^7*(1+x)).
%F 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
%F 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
%t 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 *)
%o (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
%Y Cf. A019298, A053494.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Jan 15 2000; definition revised Jul 06 2014
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