A053494 Number of symmetric 5 X 5 matrices of nonnegative integers with every row and column adding to n.

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

Offset

0,2

References

R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986; see Prop. 4.6.21, p. 235, G_5(lambda).

Links

Colin Barker, Table of n, a(n) for n = 0..1000

R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]

R. P. Stanley, Magic labelings of graphs, symmetric magic squares,..., Duke Math. J. 43 (3) (1976) 511-531, F_5(x) in Section 5.

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).

Formula

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

Mathematica

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 *)

Prog

(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

Crossrefs

Cf. A019298, A053493.

Sequence in context: A252826 A042306 A329300 * A054938 A161526 A161933

Adjacent sequences: A053491 A053492 A053493 * A053495 A053496 A053497

Keyword

nonn,easy

Author

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

Status

approved