A269742 -id:A269742 - cp's OEIS frontend

A000681 Number of n X n matrices with nonnegative entries and every row and column sum 2.

1, 1, 3, 21, 282, 6210, 202410, 9135630, 545007960, 41514583320, 3930730108200, 452785322266200, 62347376347779600, 10112899541133589200, 1908371363842760216400, 414517594539154672566000, 102681435747106627787376000, 28772944645196614863048048000

Offset: 0

N. J. A. Sloane, Simon Plouffe

Or, number of labeled 2-regular pseudodigraphs (multiple arcs and loops allowed) of order n.

Also, number of permutations of the multiset {1^2,2^2,...,n^2} with the descent set consisting of multiples of 2. - Max Alekseyev, Apr 28 2014

			G.f. = 1 + x + 3*x^2 + 21*x^3 + 282*x^4 + 6210*x^5 + 202410*x^6 + 9135630*x^7 + ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 125, #25, a_n.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983, section 3.5.10.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1982.
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, Cambridge, Vol. 2, 1999; see Cor. 5.5.11 (a).
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.
C. B. Tompkins, Methods of successive restrictions in computational problems involving discrete variables. 1963, Proc. Sympos. Appl. Math., Vol. XV pp. 95-106; Amer. Math. Soc., Providence, R.I.

Alois P. Heinz, Table of n, a(n) for n = 0..250 (first 49 terms from R. W. Robinson)
H. Anand, V. C. Dumir and H. Gupta, A combinatorial distribution problem, Duke Math. J., 33 (1996), 757-769.
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.
S. Cockburn and J. Lesperance, Deranged socks, Mathematics Magazine, 86 (2013), 97-109.
Ira Gessel, Enumerative applications of symmetric functions, Séminaire Lotharingien de Combinatoire, B17a (1987), 17 pp.
William George Griffiths, On Integer Solutions to Linear Equations, Annals of Combinatorics 12:1 (2008), pp. 53-70.
Rui-Li Liu, Feng-Zhen Zhao, New Sufficient Conditions for Log-Balancedness, With Applications to Combinatorial Sequences, J. Int. Seq., Vol. 21 (2018), Article 18.5.7.
Richard J. Mathar, 2-regular Digraphs of the Lovelock Lagrangian, arXiv:1903.12477 [math.GM], 2019.
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

Column k=2 of A257493.
Row sums of A269742 and A307804.
Row and column sums equal s: A000142 (s=1), A001500 (s=3), A172806 (s=4), A172862 (s=5), A172894 (s=6), A172919 (s=7), A172944 (s=8), A172958 (s=9).
Cf. A001499, A005650, A123544.

A000681 := proc(n)
    coeftayl( exp(x/2)/sqrt(1-x),x=0,n) ;
    %*(n!)^2 ;
end proc:
seq(A000681(n),n=0..10) ; # R. J. Mathar, May 01 2019

a[n_] := Sum[ ((2*i)!*n!^2) / (2^i*(i!^2*(n - i)!)), {i, 0, n}]/2^n; Table[ a[n], {n, 0, 17}] (* Jean-François Alcover, Dec 08 2011 *)
a[n_] := n!*HypergeometricPFQ[{1/2, -n}, {}, -2]/2^n; Table[a[n], {n, 0, 17}] (* Jean-François Alcover, Aug 08 2012 *)

Vec( serlaplace(serlaplace( exp(x/2)/sqrt(1-x) )) ) /* Max Alekseyev, Apr 28 2014 */

from sage.combinat.integer_matrices import IntegerMatrices
def a(n): return IntegerMatrices([2]*n, [2]*n).cardinality() # Ralf Stephan, Mar 02 2014

Sum_{n >= 0} a(n) x^n / n!^2 = exp(x/2) / sqrt(1-x).
D-finite with recurrence a(n) = n^2*a(n-1) - (1/2)*n*(n-1)^2*a(n-2).
a(n) is asymptotic to c/sqrt(n)*(n!)^2 where c=0.93019... - Benoit Cloitre, Jun 25 2004
a(n) = sum(i=0..n, 2^(i-2*n) * C(n, i)^2 * (2*n-2*i)! * i! ).
a(n) = 2^(-n) * sum(i=0..n, ((n!)^2*(2*i)!) / ((i!)^2*((n-i)!*2^i)) ). - Shanzhen Gao, Nov 05 2007
In Cloitre's formula is c = exp(1/2)/sqrt(Pi) = 0.9301913671026328586. - Vaclav Kotesovec, Aug 12 2013
With c as used above by Cloitre and Kotesovec, a(n) is asymptotic to c/sqrt(n)*(n!)^2 * (1 + 2/(16*n) + 50/(16*n)^2 + 1100/(16*n)^3 + 32438/(16*n)^4 + 1185660/(16*n)^5 + 50498228/(16*n)^6 + 2438464600/(16*n)^7 + 131323987366/(16*n)^8 + 7782036656108/(16*n)^9 + 501905392385436/(16*n)^10 + ...). - Jon E. Schoenfield, Mar 03 2014
E.g.f.: 2/((2-x)*W(0)), where W(k) = 1 - (2*k+1)*x/(2-x-2*(k+1)*x/W(k+1)); (continued fraction). - Sergei N. Gladkovskii, Nov 25 2014

More terms from David W. Wilson

A260585 Number of ways to place 2n rooks on an n X n board, with 2 rooks in each row and each column, multiple rooks in a cell allowed, and exactly 2 rooks below the main diagonal.

Original entry on oeis.org

1, 11, 72, 367, 1630, 6680, 26082, 98870, 368045, 1354850, 4953503, 18035279, 65499031, 237511321, 860471110, 3115667369, 11277816388, 40814611818, 147692103728, 534404499040, 1933597628291, 6996040095316, 25312367524557, 91581960107817, 331348634005165

Offset: 2

Jeffrey Davis, Jul 29 2015

a(n) is the number of minimal multiplex juggling patterns of period n using exactly 2 balls when we can catch/throw up to 2 balls at a time. (Minimal in the sense that each of the n throws is between 0 and n-1.)

Colin Barker, Table of n, a(n) for n = 2..1000
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.
Index entries for linear recurrences with constant coefficients, signature (12,-59,155,-236,209,-100,20).

Column k=2 of A269742.

Cf. A260575, A260582, A260583, A260584, A260727.

CoefficientList[Series[-(5*x^4 - 3*x^3 - x^2 - x + 1)/(20*x^7 - 100*x^6 + 209*x^5 - 236*x^4 + 155*x^3 - 59*x^2 + 12*x - 1), {x, 0, 30}], x] (* Wesley Ivan Hurt, Aug 16 2015 *)

Vec(-(5*x^6 - 3*x^5 - x^4 - x^3 + x^2)/(20*x^7 - 100*x^6 + 209*x^5 - 236*x^4 + 155*x^3 - 59*x^2 + 12*x - 1) + O(x^40)) \\ Michel Marcus, Aug 17 2015

G.f.: -x^2*(5*x^4-3*x^3-x^2-x+1)/((1-5*x+5*x^2)*(2*x-1)^2*(x-1)^3).
a(n) = 12*a(n-1) - 59*a(n-2) + 155*a(n-3) - 236*a(n-4) + 209*a(n-5) - 100*a(n-6) + 20*a(n-7). - Wesley Ivan Hurt, Jan 01 2024
a(n) = (n+2)*(n-1)/2-2^n*(1+3*n/2)+2*A030191(n)-5*A030191(n-1). - R. J. Mathar, Aug 26 2025

A260575 Number of ways to place 2n rooks on n X n board, 2 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

4, 114, 1492, 13992, 109538, 769632, 5050616, 31702275, 193204684, 1154354559, 6805263818, 39756392269, 230829718918, 1334626765852, 7694795830792, 44279453377166, 254475676808510, 1461211112505546, 8385454709982584, 48102877501302765, 275868835046218560

Offset: 3

Steve Butler, Jul 29 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 2 balls at a time. (Minimal in the sense that the throws 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=3 of A269742.

Cf. A260582, A260583, A260584, A260585, A260727.

Rest[Rest[Rest[CoefficientList[Series[-(625 x^12 - 2206 x^11 + 4397 x^10 - 6648 x^9 + 7058 x^8 - 4674 x^7 + 1748 x^6 - 300 x^5 + 2 x^4 + 4 x^3)/(2600 x^13 - 21100 x^12 + 77590 x^11 - 171025 x^10 + 251874 x^9 - 261466 x^8 + 196626 x^7 - 108337 x^6 + 43682 x^5 - 12713 x^4 + 2592 x^3 - 350 x^2 + 28 x - 1), {x, 0, 33}], x]]]] (* Vincenzo Librandi, Jul 30 2015 *)

G.f.: -(625*x^12 - 2206*x^11 + 4397*x^10 - 6648*x^9 + 7058*x^8 - 4674*x^7 + 1748*x^6 - 300*x^5 + 2*x^4 + 4*x^3)/(2600*x^13 - 21100*x^12 + 77590*x^11 - 171025*x^10 + 251874*x^9 - 261466*x^8 + 196626*x^7 - 108337*x^6 + 43682*x^5 - 12713*x^4 + 2592*x^3 - 350*x^2 + 28*x - 1).

A260582 Number of ways to place 2n rooks on n X n board, 2 rooks in each row and each column, multiple rooks in a cell allowed, and exactly 4 rooks below the main diagonal.

Original entry on oeis.org

1, 72, 2438, 48965, 727982, 9002669, 98831244, 1001534339, 9604385112, 88600727292, 795108048465, 6995452987296, 60672964077315, 520801298224219, 4436874672072459, 37592602817393616, 317246106027904761, 2669508900483670024, 22415690107381454687

Offset: 3

Scarlitte Ponce, 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 2 balls at a time. (Minimal in the sense that the number of throws is 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 A269742.

Cf. A260575, A260583, A260584, A260585, A260727.

CoefficientList[Series[-(4584825 x^18 - 32402639 x^17 + 116197885 x^16 - 276109240 x^15 + 466638686 x^14 - 565360388 x^13 + 479478061 x^12 - 263101580 x^11 + 64737485 x^10 + 27721713 x^9 - 36043190 x^8 + 18319939 x^7 - 5637417 x^6 + 1094626 x^5 - 124221 x^4 + 5875 x^3 + 171 x^2 - 14 x - 1)/(19266000 x^22 - 234624000 x^21 + 1345258600 x^20 - 4829459800 x^19 + 12177772645 x^18 - 22934336190  x^17 + 33487611783 x^16 - 38844575208 x^15 + 36384232939 x^14 - 27820436326 x^13 + 17485343731 x^12 - 9066508172 x^11 + 3881838842 x^10 - 1369857572 x^9 + 396588486 x^8 - 93458208 x^7 + 17715207 x^6 - 2654590 x^5 + 306583 x^4 - 26260 x^3 + 1567 x^2 - 58 x + 1), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 19 2015 *)

G.f.: -(4584825*x^21 - 32402639*x^20 + 116197885*x^19 - 276109240*x^18 + 466638686*x^17 - 565360388*x^16 + 479478061*x^15 - 263101580*x^14 + 64737485*x^13 + 27721713*x^12 - 36043190*x^11 + 18319939*x^10 - 5637417*x^9 + 1094626*x^8 - 124221*x^7 + 5875*x^6 + 171*x^5 - 14*x^4 - x^3)/(19266000*x^22 - 234624000*x^21 + 1345258600*x^20 - 4829459800*x^19 + 12177772645*x^18 - 22934336190*x^17 + 33487611783*x^16 - 38844575208*x^15 + 36384232939*x^14 - 27820436326*x^13 + 17485343731*x^12 - 9066508172*x^11 + 3881838842*x^10 - 1369857572*x^9 + 396588486*x^8 - 93458208*x^7 + 17715207*x^6 - 2654590*x^5 + 306583*x^4 - 26260*x^3 + 1567*x^2 - 58*x + 1).

A269743 Triangle of generalized Eulerian numbers T(n,k) = _3 read by rows, n >= 1, 0 <= k <= 3*(n-1).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 4, 11, 23, 11, 4, 1, 1, 11, 72, 325, 595, 595, 325, 72, 11, 1, 1, 26, 367, 3368, 14679, 34679, 46800, 34679, 14679, 3368, 367, 26, 1, 1, 57, 1630, 28819, 253247, 1212440, 3382133, 5588593, 5588593, 3382133, 1212440, 253247, 28819, 1630, 57, 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 3 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,
1,4,11,23,11,4,1,
1,11,72,325,595,595,325,72,11,1,
...

Andrew Howroyd, Table of n, a(n) for n = 1..925 (first 25 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 A001500.
Columns k=0..4 are A000012, A000295, A260585, A260727, A260583.
Cf. A008292, A269742, A269744.

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

Terms a(23) and beyond from Andrew Howroyd, Feb 22 2020

A332729 Number of nonnegative integer n X n matrices with row and column sums 2 and sum of entries below the main diagonal n - 1.

Original entry on oeis.org

1, 1, 11, 114, 2438, 73120, 3107640, 175157568, 12683256458, 1146644890542, 126603925984322, 16764298017398342, 2622239904802101734, 478350006311298468126, 100655530463934465864626, 24200010145455307873369888, 6592700960401481917596215614, 2020180735699844322843722782402

Offset: 1

Andrew Howroyd, Feb 22 2020

nonn

			The a(2) = 1 matrix is:
   [1 1]
   [1 1]

Jean-François Alcover, Table of n, a(n) for n = 1..40 (all terms from Andrew Howroyd)

Central coefficients of A269742.

A269742 = Cases[Import["https://oeis.org/A269742/b269742.txt", "Table"], {, }][[All, 2]];
a[n_] := A269742[[n^2 - n + 1]];
Array[a, 40] (* Jean-François Alcover, Feb 24 2020 *)

cp's OEIS Frontend

A000681 Number of n X n matrices with nonnegative entries and every row and column sum 2.

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A260585 Number of ways to place 2n rooks on an n X n board, with 2 rooks in each row and each column, multiple rooks in a cell allowed, and exactly 2 rooks below the main diagonal.

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A260575 Number of ways to place 2n rooks on n X n board, 2 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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A260582 Number of ways to place 2n rooks on n X n board, 2 rooks in each row and each column, multiple rooks in a cell allowed, and exactly 4 rooks below the main diagonal.

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A269743 Triangle of generalized Eulerian numbers T(n,k) = _3 read by rows, n >= 1, 0 <= k <= 3*(n-1).

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A332729 Number of nonnegative integer n X n matrices with row and column sums 2 and sum of entries below the main diagonal n - 1.

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