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A053494
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Number of symmetric 5 X 5 matrices of nonnegative integers with every row and column adding to n.
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5
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1, 26, 348, 2698, 14751, 62781, 222190, 681460, 1865715, 4655535, 10756921, 23290026, 47700173, 93104473, 174248451, 314246511, 548380980, 929209095, 1533389605, 2470568045, 3894914166, 6019752376, 9136114923, 13635769173, 20039850376, 29033765566
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OFFSET
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0,2
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REFERENCES
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R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986; see Prop. 4.6.21, p. 235, G_5(lambda).
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (5,-4,-20,40,16,-100,44,110,-110,-44,100,-16,-40,20,4,-5,1).
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FORMULA
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G.f.: (1 + 21*x + 222*x^2 + 1082*x^3 + 3133*x^4 + 5722*x^5 + 7013*x^6 + 5722*x^7 + 3133*x^8 + 1082*x^9 + 222*x^10 + 21*x^11 + x^12) / ((1-x)^11*(1+x)^6).
a(n) = (189*(59981+5555*(-1)^n) + 18*(2345165+65331*(-1)^n)*n + (76615494+689850*(-1)^n)*n^2 + 40*(2138179+6237*(-1)^n)*n^3 + (63277966+47250*(-1)^n)*n^4 + 1260*(25421+3*(-1)^n)*n^5 + 11171664*n^6 + 2644080*n^7 + 405954*n^8 + 36500*n^9 + 1460*n^10) / 12386304. - Colin Barker, Jan 14 2017
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MATHEMATICA
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CoefficientList[Series[(1+21x+222x^2+1082x^3+3133x^4+5722x^5+7013x^6+5722x^7+3133x^8+1082x^9+222x^10+21x^11+x^12)/((1-x)^11(1+x)^6), {x, 0, 30}], x] (* or *) LinearRecurrence[ {5, -4, -20, 40, 16, -100, 44, 110, -110, -44, 100, -16, -40, 20, 4, -5, 1}, {1, 26, 348, 2698, 14751, 62781, 222190, 681460, 1865715, 4655535, 10756921, 23290026, 47700173, 93104473, 174248451, 314246511, 548380980}, 30] (* Harvey P. Dale, Mar 05 2023 *)
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PROG
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(PARI) Vec((1 + 21*x + 222*x^2 + 1082*x^3 + 3133*x^4 + 5722*x^5 + 7013*x^6 + 5722*x^7 + 3133*x^8 + 1082*x^9 + 222*x^10 + 21*x^11 + x^12) / ((1 - x)^11*(1 + x)^6) + O(x^30)) \\ Colin Barker, Jan 14 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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