0,3

a(n)=6^{-n}\dsum\limits_{alpha =0}^{n}\dsum\limits_{beta =0}^{n-alpha }\sum_{alpha +beta +\gamma =n}frac{2^{alpha }3^{beta }(n!)^{2} (-2beta +3n-3alpha )!}{alpha !beta !(n-alpha -beta )!^{2}6^{(n-alpha -beta )}}. - Shanzhen Gao (sgao2(AT)fau.edu), Nov 05 2007

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 125, Problem 25(4), b_n (but beware errors).

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.

Goulden, I. P.; Jackson, D. M.; Reilly, J. W.; The Hammond series of a symmetric function and its application to $P$-recursiveness. SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 179-193.

M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements. Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970.

Index entries for sequences related to magic squares

E.g.f. y(x) = Sum_{n >= 0} a(n)*x^n/(n!)^2 satisfies differential equation 81*x^5*(x^4 - x^2 + x + 4)*diff(y(x), x, x, x, x) + 324*x^4*(x^4 - x^2 + x + 4)*diff(y(x), x, x, x) - 9*x*(x^10 - 4*x^9 + 22*x^8 - 8*x^7 - 22*x^6 + 8*x^5 + 106*x^4 + 234*x^3 + 48*x^2 (cont.)

(cont.) - 320*x + 64)*diff(y(x), x, x) - 9*(x^10 - 4*x^9 + 22*x^8 - 8*x^7 - 4*x^6 + 8*x^5 + 88*x^4 + 252*x^3 + 120*x^2 - 320*x + 64)*diff(y(x), x) + (x^11 - 7*x^10 + 30*x^9 - 16*x^8 - 43*x^7 + 51*x^6 + 238*x^5 + 630*x^4 + 36*x^3 - 1944*x^2 - 1152*x + 576)*y(x) = 0.

Recurrence: a(n) = n!*v(n) where v(n) = 1/(576*n)*(( - 198*n^9 + 8712*n^8 - 165175*n^7 + 1764196*n^6 - 11643772*n^5 + 48965728*n^4 - 130257475*n^3 + 209370724*n^2 - 182126340*n + 64083600)*v(n - 8) + (36*n^10 - 1944*n^9 + 45884*n^8 - 621504*n^7 + 5330892*n^6 - 30123576*n^5 + 112954596*n^4 - 275612976*n^3 + 415021552*n^2 - 343920960*n + 116928000)*v(n - 9) (cont.)

(cont.) + ( - 9*n^11 + 585*n^10 - 16800*n^9 + 280800*n^8 - 3027357*n^7 + 22034565*n^6 - 110039130*n^5 + 375129450*n^4 - 849926784*n^3 + 1208298600*n^2 - 958439520*n + 315705600)*v(n - 10) + ( - 7*n^10 + 385*n^9 - 9240*n^8 + 127050*n^7 - 1104411*n^6 + 6314385*n^5 - 23918510*n^4 + 58866500*n^3 - 89275032*n^2 + 74! 400480*n - 25401600)*v(n - 11) (cont.)

(cont.) + ( - 81*n^7 + 1944*n^6 - 20232*n^5 + 115578*n^4 - 383283*n^3 + 724230*n^2 - 708372*n + 270216)*v(n - 4) + ( - 72*n^6 + 1440*n^5 - 10890*n^4 + 40500*n^3 - 78678*n^2 + 75780*n - 28080)*v(n - 5) + (81*n^9 - 3321*n^8 + 59004*n^7 - 594054*n^6 + 3718687*n^5 - 14927199*n^4 + 38152096*n^3 - 59311746*n^2 + 50236612*n - 17330160)*v(n - 6) + (72*n^8 - 2520*n^7 + 37347*n^6 - 304479*n^5 + 1484133*n^4 - 4394565*n^3 + 7642248*n^2 - 7039116*n (cont.)

(cont.) + 2576880)*v(n - 7) + (n^11 - 66*n^10 + 1925*n^9 - 32670*n^8 + 357423*n^7 - 2637558*n^6 + 13339535*n^5 - 45995730*n^4 + 105258076*n^3 - 150917976*n^2 + 120543840*n - 39916800)*v(n - 12) + (2880*n^2 - 5760*n + 3456)*v(n - 1) + (324*n^5 - 3564*n^4 + 14148*n^3 - 26028*n^2 + 21312*n - 6192)*v(n - 2) + (81*n^6 - 1377*n^5 + 7209*n^4 - 13203*n^3 - 3402*n^2 + 32076*n - 21384)*v(n - 3)).

Cf. A000681.

Sequence in context: A133218 A073352 A099122 this_sequence A054751 A006592 A077122

Adjacent sequences: A001497 A001498 A001499 this_sequence A001501 A001502 A001503

nonn,easy

njas

More terms and formulae from Vladeta Jovovic (vladeta(AT)Eunet.yu), Mar 26 2001