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A001500
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Number of stochastic matrices of integers: n X n arrays of nonnegative integers with all row and column sums equal to 3.
(Formerly M3689 N1507)
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1
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1, 1, 4, 55, 2008, 153040, 20933840, 4662857360, 1579060246400, 772200774683520, 523853880779443200, 477360556805016931200, 569060910292172349004800, 868071731152923490921728000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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
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REFERENCES
| 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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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.
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LINKS
| Index entries for sequences related to magic squares
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FORMULA
| 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)).
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CROSSREFS
| Cf. A000681.
Sequence in context: A195634 A206384 A099122 * A054751 A189873 A143650
Adjacent sequences: A001497 A001498 A001499 * A001501 A001502 A001503
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms and formulae from Vladeta Jovovic (vladeta(AT)eunet.rs), Mar 26 2001
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