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A001496
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Number of 4 X 4 matrices with nonnegative integer entries and row and column sums equal to n.
(Formerly M5158 N2240)
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20
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1, 24, 282, 2008, 10147, 40176, 132724, 381424, 981541, 2309384, 5045326, 10356424, 20158151, 37478624, 66952936, 115479776, 193077449, 313981688, 498033282, 772409528, 1173759851, 1750812624, 2567527260, 3706873040
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OFFSET
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0,2
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COMMENTS
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Number of 4 X 4 stochastic matrices of integers.
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REFERENCES
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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.
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
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FORMULA
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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].
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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See A002721 for a 3-dimensional analog.
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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