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%I M5158 N2240 #61 Jan 23 2019 12:36:25
%S 1,24,282,2008,10147,40176,132724,381424,981541,2309384,5045326,
%T 10356424,20158151,37478624,66952936,115479776,193077449,313981688,
%U 498033282,772409528,1173759851,1750812624,2567527260,3706873040
%N Number of 4 X 4 matrices with nonnegative integer entries and row and column sums equal to n.
%C Number of 4 X 4 stochastic matrices of integers.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 124, #25, Q(4,r).
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, pages 233-234.
%D 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.
%H T. D. Noe, <a href="/A001496/b001496.txt">Table of n, a(n) for n = 0..1000</a>
%H A. G. Bell, <a href="http://dx.doi.org/10.1093/comjnl/13.3.278">Partitioning integers in n dimensions</a>, The Computer Journal, 13 (1970), 278-283.
%H Brian Conrey and Alex Gamburd, <a href="http://dx.doi.org/10.1016/j.jnt.2005.01.006">Pseudomoments of the Riemann zeta-function and pseudomagic squares</a>, Journal of Number Theory, Volume 117, Issue 2, April 2006, Pages 263-278.
%H I. J. Good, <a href="http://www.jstor.org/stable/2958586">On the application of symmetric Dirichlet distributions and their mixtures to contingency tables</a>, Ann. Statist. 4 (1976), no. 6, 1159-1189.
%H I. J. Good, <a href="/A001496/a001496_1.pdf">On the application of symmetric Dirichlet distributions and contingency tables</a>, pp. 1178-1179. (Annotated scanned copy)
%H D. M. Jackson and G. H. J. van Rees, <a href="http://dx.doi.org/10.1137/0204040">The enumeration of generalized double stochastic nonnegative integer square matrices</a>, SIAM J. Comput., 4 (1975), 474-477.
%H D. M. Jackson & G. H. J. van Rees, <a href="/A002817/a002817.pdf">The enumeration of generalized double stochastic nonnegative integer square matrices</a>, SIAM J. Comput., 4.4 (1975), 474-477. (Annotated scanned copy)
%H M. L. Stein and P. R. Stein, <a href="/A001496/a001496.pdf">Enumeration of Stochastic Matrices with Integer Elements</a>, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]
%H J. N. van Rijn, F. W. Takes, J. K. Vis, <a href="http://liacs.leidenuniv.nl/~rijnjnvan/pdf/pub/bnaic2018.pdf">Computing and Predicting Winning Hands in the Trick-Taking Game of Klaverjas</a>, 30th Benelux Conference on Artificial Intelligence (BNAIC 2018), 's-Hertogenbosch, the Netherlands.
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F G.f.: (1+14*x+87*x^2+148*x^3+87*x^4+14*x^5+x^6)/(1-x)^10.
%F a(n) = binomial(n + 3, 3) + 20*binomial(n + 4, 5) + 152*binomial(n + 5, 7) + 352*binomial(n + 6, 9). [Equivalent to a formula given by Bell].
%t CoefficientList[Series[(1 + 14*x + 87*x^2 + 148*x^3 + 87*x^4 + 14*x^5 + x^6)/(1 - x)^10, {x, 0, 30}], x] (* _Wesley Ivan Hurt_, Jan 24 2017 *)
%t LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{1,24,282,2008,10147,40176,132724,381424,981541,2309384},30] (* _Harvey P. Dale_, Jul 12 2017 *)
%o (PARI) x='x+O('x^99); Vec((1+14*x+87*x^2+148*x^3+87*x^4+14*x^5+x^6)/(1-x)^10) \\ _Altug Alkan_, Apr 17 2016
%Y Cf. A002817, A003438, A019298.
%Y See A002721 for a 3-dimensional analog.
%Y Row n=4 of A257493.
%K nonn,easy,nice
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Vladeta Jovovic_, Feb 06 2000
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