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A000681 Number of n X n matrices with nonnegative entries and every row and column sum 2.
(Formerly M3084 N1250)
15
1, 1, 3, 21, 282, 6210, 202410, 9135630, 545007960, 41514583320, 3930730108200, 452785322266200, 62347376347779600, 10112899541133589200, 1908371363842760216400, 414517594539154672566000, 102681435747106627787376000, 28772944645196614863048048000 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
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
REFERENCES
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.
LINKS
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]
FORMULA
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
EXAMPLE
G.f. = 1 + x + 3*x^2 + 21*x^3 + 282*x^4 + 6210*x^5 + 202410*x^6 + 9135630*x^7 + ...
MAPLE
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
MATHEMATICA
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 *)
PROG
(Sage) from sage.combinat.integer_matrices import IntegerMatrices
def a(n): return IntegerMatrices([2]*n, [2]*n).cardinality() # Ralf Stephan, Mar 02 2014
(PARI) Vec( serlaplace(serlaplace( exp(x/2)/sqrt(1-x) )) ) /* Max Alekseyev, Apr 28 2014 */
CROSSREFS
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).
Sequence in context: A130032 A174967 A126461 * A222035 A361214 A171201
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
More terms from David W. Wilson
STATUS
approved

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