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

Author

N. J. A. Sloane, Jan 15 2000

Keywords

• nonn
• easy

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• R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]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

Crossrefs

Cf. A019298, A053493.

Programs

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 *)

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

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

Extensions

Revised definition, Jul 06 2014