A005466 -id:A005466 - cp's OEIS frontend

A003438 Number of 5 X 5 matrices with nonnegative integer entries and row and column sums equal to n.

1, 120, 6210, 153040, 2224955, 22069251, 164176640, 976395820, 4855258305, 20856798285, 79315936751, 272095118010, 854560160105, 2486299719645, 6765755480415, 17356306529251, 42250330784180, 98137852369965

Offset: 0

N. J. A. Sloane

Number of 5 X 5 stochastic matrices of integers.

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.
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, p. 234.

T. D. Noe, Table of n, a(n) for n = 0..1000
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]
Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Cf. A002817, A001496, A019298.
Cf. A001496, A005466, A058391.
Row n=5 of A257493.

CoefficientList[Series[(1+103x+4306x^2+63110x^3+388615x^4+1115068x^5+ 1575669x^6+1115068x^7+388615x^8+63110x^9+4306x^10+103x^11+x^12)/ (1-x)^17,{x,0,30}],x] (* Harvey P. Dale, Aug 17 2013 *)

G.f.: (1 + 103*x + 4306*x^2 + 63110*x^3 + 388615*x^4 + 1115068*x^5 + 1575669*x^6 + 1115068*x^7 + 388615*x^8 + 63110*x^9 + 4306*x^10 + 103*x^11 + x^12)/(1-x)^17.

a(n) = Sum_{j=0..6} A005466(j) * binomial(4+j+n, 4+2*j). - Andrew Howroyd, Apr 09 2020

More terms from Vladeta Jovovic, Feb 06 2000

A005467 6 X 6 stochastic matrices of integers.

Original entry on oeis.org

1, 714, 196677, 18941310, 809451144, 17914693608, 223688514048, 1633645276848, 6907466271384, 15642484909560, 14666561365176

Offset: 0

N. J. A. Sloane

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.

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)
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]

Cf. A005466.

a(10) corrected by Andrew Howroyd, Apr 09 2020

A259473 Irregular triangle read by rows of coefficients arising in the enumeration of doubly stochastic matrices of integers, n >= 1, 0 <= k <= (n-1)*(n-2).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 14, 87, 148, 87, 14, 1, 1, 103, 4306, 63110, 388615, 1115068, 1575669, 1115068, 388615, 63110, 4306, 103, 1, 1, 694, 184015, 15902580, 567296265, 9816969306, 91422589980, 490333468494, 1583419977390, 3166404385990, 3982599815746, 3166404385990

Offset: 1

N. J. A. Sloane, Jul 03 2015

The n-th row of A257493 is a polynomial of degree (n-1)^2. This triangle gives the coefficients of the numerator of the generating functions for A257493 with denominators being (1-x)^(1+(n-1)^2). - Andrew Howroyd, Apr 11 2020

			Triangle begins:
  1;
  1;
  1,1,1;
  1,14,87,148,87,14,1;
  1,103,4306,63110,388615,1115068,1575669,1115068,388615,63110,4306,103,1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..177 (rows 1..9)
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, doi:10.1137/0204040.
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)

Row sums are A037302.

Cf. A005466, A005467, A257493.

T(n,k) = Sum_{i=0..k} A257493(n, k-i)*(-1)^i*binomial(1+(n-1)^2,i). - Andrew Howroyd, Apr 11 2020

a(1)=1 prepended and terms a(26) and beyond from Andrew Howroyd, Apr 11 2020

cp's OEIS Frontend

A003438 Number of 5 X 5 matrices with nonnegative integer entries and row and column sums equal to n.

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A005467 6 X 6 stochastic matrices of integers.

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A259473 Irregular triangle read by rows of coefficients arising in the enumeration of doubly stochastic matrices of integers, n >= 1, 0 <= k <= (n-1)*(n-2).

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