A269743 -id:A269743 - cp's OEIS frontend

A001500 Number of stochastic matrices of integers: n X n arrays of nonnegative integers with all row and column sums equal to 3.

1, 1, 4, 55, 2008, 153040, 20933840, 4662857360, 1579060246400, 772200774683520, 523853880779443200, 477360556805016931200, 569060910292172349004800, 868071731152923490921728000, 1663043727673392444887284377600, 3937477620391471128913917360384000

Offset: 0

N. J. A. Sloane

Also, number of bicubical multigraphs on 2n labeled nodes of two colors [Read, 1958, 1971]. - N. J. A. Sloane, Sep 09 2014

			a(2) = 4 with: [0 3]    [1 2]    [2 1]    [3 0]
               [3 0],   [2 1],   [1 2],   [0 3]. - _Bernard Schott_, Oct 15 2019

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.
R. C. Read, Some Enumeration Problems in Graph Theory. Ph.D. Dissertation, Department of Mathematics, Univ. London, 1958.
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.

Alois P. Heinz, Table of n, a(n) for n = 0..180
Esther M. Banaian, Generalized Eulerian Numbers and Multiplex Juggling Sequences, (2016). All College Thesis Program. Paper 24.
E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.
Petter Brändén, Jonathan Leake, Igor Pak, Lower bounds for contingency tables via Lorentzian polynomials, arXiv:2008.05907 [math.CO], 2020.
I. P. Goulden, D. M. Jackson, and J. W. Reilly, The Hammond series of a symmetric function and its application to P-recursiveness, SIAM J. Algebraic Discrete Methods 4 (1983), no. 2, 179-193.
R. C. Read, Letter to N. J. A. Sloane, Feb 04 1971 (gives initial terms of this sequence, although there are errors)
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 sequences related to magic squares

Row sums of A269743 and of A344379.
Column k=3 of A257493.
Cf. A000681, A246970.

a[n_] := 6^(-n) Sum[2^j 3^k n!^2 (3n - 2k - 3j)!/(j! k! (n - j - k)!^2 * 6^(n - j - k)), {j, 0, n}, {k, 0, n - j}];
a /@ Range[0, 15] (* Jean-François Alcover, Oct 15 2019, after Shanzhen Gao *)

From Vladeta Jovovic, Mar 26 2001: (Start)
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)*(d^4/dx^4)y(x) + 324*x^4*(x^4 - x^2 + x + 4)*(d^3/dx^3)y(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 - 320*x + 64)*(d^2/dx^2)y(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)*(d/dx)y(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) + (-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 + 74400480*n - 25401600)*v(n - 11) + (-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 + 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)). (End)
a(n) = 6^(-n) * Sum_{ alpha = 0..n, beta = 0..n-alpha } (2^alpha*3^beta*(n!)^2*(-2*beta+3*n-3*alpha)!)/(alpha!*beta!*(n-alpha-beta)!^2*6^(n-alpha-beta)). - Shanzhen Gao, Nov 05 2007
a(n) ~ sqrt(Pi) * 3^(n + 1/2) * n^(3*n + 1/2) / (2^(2*n - 1/2) * exp(3*n - 2)). - Vaclav Kotesovec, Oct 15 2019

More terms from Vladeta Jovovic, Mar 26 2001

A260583 Number of ways to place 3n rooks on an n X n board, 3 rooks in each row and each column, multiple rooks in an allowed cell, and exactly 4 rooks below the main diagonal.

Original entry on oeis.org

11, 595, 14679, 253247, 3564803, 44226950, 505572550, 5473391465, 57122380066, 581477852342, 5819301681925, 57564437594318, 564911137682637, 5513703983635512, 53616132982114742, 520057429817203110, 5035740328012627416, 48704838658567681135

Offset: 3

Esther Banaian, Jul 29 2015

nonn

a(n) is the number of minimal multiplex juggling patterns of period n using exactly 4 balls when we can catch/throw up to 3 balls at a time. (Minimal in the sense that the throw heights are between 0 and n-1.)

Colin Barker, Table of n, a(n) for n = 3..1000
E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.

Column k=4 of A269743.

Cf. A260575, A260582, A260584, A260585, A260727.

G.f.: -(24558000*x^24 - 221169800*x^23 + 1030045255*x^22 - 3270869391*x^21 + 7705144467*x^20 - 13843184523*x^19 + 19209151138*x^18 - 20800159606*x^17 + 17768204859*x^16 - 12126221923*x^15 + 6718636422*x^14 - 3086566305*x^13 + 1204914514*x^12 - 407103232*x^11 + 118646908*x^10 - 28836372*x^9 + 5505383*x^8 - 758705*x^7 + 65305*x^6 - 2162*x^5 - 131*x^4 + 11*x^3)/(133600000*x^25 - 1875920000*x^24 + 12500686000*x^23 - 52604444000*x^22 + 156920670600*x^21 - 353103818000*x^20 + 622718972395*x^19 - 882777307660*x^18 + 1023713051333*x^17 - 983132187597*x^16 + 788634518440*x^15 - 531447118763*x^14 + 301890662895*x^13 - 144761728498*x^12 + 58568440406*x^11 - 19945788669*x^10 + 5692551701*x^9 - 1352405718*x^8 + 264899104*x^7 - 42210805*x^6 + 5370925*x^5 - 531418*x^4 + 39303*x^3 - 2039*x^2 + 66*x - 1).

A260727 Number of ways to place 3n rooks on an n X n board, with 3 rooks in each row and each column, multiple rooks in a cell allowed, and exactly 3 rooks below the main diagonal.

Original entry on oeis.org

1, 23, 325, 3368, 28819, 218788, 1539399, 10314315, 66953292, 425761614, 2671506918, 16618186770, 102796975770, 633596982417, 3896224129259, 23924104985984, 146764696175937, 899809941054468, 5514653407814317, 33789681789605283, 207007665004469906

Offset: 2

Chris Cox, Jul 30 2015

a(n) is the number of minimal multiplex juggling patterns of period n using exactly 3 balls when we can catch/throw up to 3 balls at a time. (Minimal in the sense that each of the n throws is between 0 and n-1, inclusive.)

Colin Barker, Table of n, a(n) for n = 2..1000
E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce, A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.

Column k=3 of A269743.

Cf. A260575, A260582, A260583, A260584, A260585.

CoefficientList[Series[-(700*x^13 - 2435*x^12 + 4558*x^11 - 7532*x^10 + 10404*x^9 - 9697*x^8 + 5545*x^7 - 1844*x^6 + 336*x^5 - 39*x^4 + 7*x^3 - x^2)/((x^2)*(4000*x^14 - 35400*x^13 + 143100*x^12 - 349910*x^11 + 577675*x^10 - 680496*x^9 + 589248*x^8 - 380592*x^7 + 184037*x^6 - 66214*x^5 + 17423*x^4 - 3246*x^3 + 404*x^2 - 30*x + 1)), {x, 0, 20}], x] (* Wesley Ivan Hurt, Jul 30 2015 *)

G.f.: -(700*x^13 - 2435*x^12 + 4558*x^11 - 7532*x^10 + 10404*x^9 - 9697*x^8 + 5545*x^7 - 1844*x^6 + 336*x^5 - 39*x^4 + 7*x^3 - x^2)/(4000*x^14 - 35400*x^13 + 143100*x^12 - 349910*x^11 + 577675*x^10 - 680496*x^9 + 589248*x^8 - 380592*x^7 + 184037*x^6 - 66214*x^5 + 17423*x^4 - 3246*x^3 + 404*x^2 - 30*x + 1).

A269742 Triangle of generalized Eulerian numbers T(n,k) = _2 read by rows, n >= 1, 0 <= k < 2*n.

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 11, 4, 1, 1, 11, 72, 114, 72, 11, 1, 1, 26, 367, 1492, 2438, 1492, 367, 26, 1, 1, 57, 1630, 13992, 48965, 73120, 48965, 13992, 1630, 57, 1, 1, 120, 6680, 109538, 727982, 2169674, 3107640, 2169674, 727982, 109538, 6680, 120, 1

Offset: 1

N. J. A. Sloane, Mar 16 2016

T(n,k) is the number of nonnegative integer n X n matrices with every row and column sum 2 and sum of entries below the main diagonal k. The case when every row and column sum is 1 is given by the Eulerian numbers (A008292). - Andrew Howroyd, Feb 22 2020

			Triangle begins:
  1;
  1, 1, 1;
  1, 4, 11, 4, 1;
  1, 11, 72, 114, 72, 11, 1;
  1, 26, 367, 1492, 2438, 1492, 367, 26, 1;
  1, 57, 1630, 13992, 48965, 73120, 48965, 13992, 1630, 57, 1;
  ...
The matrices for row n=3, k=0..2 are:
  [2 0]  [1 1]  [0 2]
  [0 2]  [1 1]  [2 0]

Andrew Howroyd, Table of n, a(n) for n = 1..1600 (first 40 rows)
Esther M. Banaian, Generalized Eulerian Numbers and Multiplex Juggling Sequences, (2016). All College Thesis Program. Paper 24.
E. Banaian, S. Butler, C. Cox, J. Davis, J. Landgraf and S. Ponce, A generalization of Eulerian numbers via rook placements, arXiv:1508.03673 [math.CO], 2015.
Andrew Howroyd, PARI Program

Row sums are A000681.
Columns k=0..4 are A000012, A000295, A260585, A260575, A260582.
Central coefficients are A332729.
Cf. A008292, A269743, A269744.

\\ See link. - Andrew Howroyd, Feb 22 2020

Terms a(26) and beyond from Andrew Howroyd, Feb 22 2020

cp's OEIS Frontend

A001500 Number of stochastic matrices of integers: n X n arrays of nonnegative integers with all row and column sums equal to 3.

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A260583 Number of ways to place 3n rooks on an n X n board, 3 rooks in each row and each column, multiple rooks in an allowed cell, and exactly 4 rooks below the main diagonal.

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A260727 Number of ways to place 3n rooks on an n X n board, with 3 rooks in each row and each column, multiple rooks in a cell allowed, and exactly 3 rooks below the main diagonal.

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A269742 Triangle of generalized Eulerian numbers T(n,k) = _2 read by rows, n >= 1, 0 <= k < 2*n.

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