A001496 Number of 4 X 4 matrices with nonnegative integer entries and row and column sums equal to n.

1, 24, 282, 2008, 10147, 40176, 132724, 381424, 981541, 2309384, 5045326, 10356424, 20158151, 37478624, 66952936, 115479776, 193077449, 313981688, 498033282, 772409528, 1173759851, 1750812624, 2567527260, 3706873040

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Number of 4 X 4 stochastic matrices of integers.

References

• L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 124, #25, Q(4,r).
• 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).
• R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, pages 233-234.
• 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.

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• A. G. Bell, Partitioning integers in n dimensions, The Computer Journal, 13 (1970), 278-283.
• Brian Conrey and Alex Gamburd, Pseudomoments of the Riemann zeta-function and pseudomagic squares, Journal of Number Theory, Volume 117, Issue 2, April 2006, Pages 263-278.
• I. J. Good, On the application of symmetric Dirichlet distributions and their mixtures to contingency tables, Ann. Statist. 4 (1976), no. 6, 1159-1189.
• I. J. Good, On the application of symmetric Dirichlet distributions and contingency tables, pp. 1178-1179. (Annotated scanned copy)
• D. M. Jackson and G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4 (1975), 474-477.
• D. M. Jackson & G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4.4 (1975), 474-477. (Annotated scanned copy)
• 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. [Annotated scanned copy]
• J. N. van Rijn, F. W. Takes, J. K. Vis, Computing and Predicting Winning Hands in the Trick-Taking Game of Klaverjas, 30th Benelux Conference on Artificial Intelligence (BNAIC 2018), 's-Hertogenbosch, the Netherlands.
• Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).

Crossrefs

Cf. A002817, A003438, A019298.

See A002721 for a 3-dimensional analog.

Row n=4 of A257493.

Programs

Mathematica

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 *)
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 *)

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

Formula

G.f.: (1+14*x+87*x^2+148*x^3+87*x^4+14*x^5+x^6)/(1-x)^10.

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].

Extensions

More terms from Vladeta Jovovic, Feb 06 2000