%I #26 Nov 01 2019 12:34:49

%S 1,2,7,34,221,1736,15584,153228,1611189,17826202,205282376,2441437708,

%T 29816628471,372314544202,4737438631001,61264426341926,

%U 803488037899349,10668478221202710,143203795004873285,1940953294927992976,26536578116407809962,365653739580163294032

%N Number of directed (or Gale-Ryser) graphical partitions: degree-vector pairs (in-degree, out-degree) for directed graphs (loops allowed) with n vertices; or possible ordered pair (row-sum, column-sum) vectors for a 0-1 matrix.

%D R. A. Brualdi, H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.

%H Andrew Howroyd, <a href="/A029894/b029894.txt">Table of n, a(n) for n = 0..30</a>

%H Peter L. Erdos, I Miklós, Z Toroczkai, <a href="http://arxiv.org/abs/1601.08224">New classes of degree sequences with fast mixing swap Markov chain sampling</a>, arXiv preprint arXiv:1601.08224 [math.CO], 2016.

%H <a href="/index/Gra#graph_part">Index entries for sequences related to graphical partitions</a>

%F Calculated using Cor. 6.3.3, Th. 6.3.6, Cor. 6.2.5 of Brualdi-Ryser.

%F a(n) = F(n, n, 0, n) where F(b, c, t, w) = Sum_{i=0..b} Sum_{j=ceiling((t+i)/w))..min(t+i, c)} F(i, j, t+i-j, w-1) for w > 0, F(b, c, 0, 0) = 1 and F(b, c, t, 0) = 0 for t > 0. - _Andrew Howroyd_, Nov 01 2019

%o (PARI) \\ see A327913 for T(n,m)

%o for(n=0, 15, print1(T(n,n), ", ")) \\ _Andrew Howroyd_, Nov 01 2019

%Y Main diagonal of A327913.

%Y Cf. A000569, A004250, A004251, A029889, A318396.

%K nonn

%O 0,2

%A torsten.sillke(AT)lhsystems.com

%E "Loops allowed" added to the definition by _Brendan McKay_, Oct 20 2015

%E a(0)=1 prepended and terms a(12) and beyond from _Andrew Howroyd_, Oct 31 2019