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%I #27 Mar 10 2023 02:25:37
%S 90,67950,90291600,154700988750,306407299538340,666569141498660400,
%T 1548539246648239560000,3776577900841430197548750,
%U 9561215418596022668009737500,24935177268489106332174087326700
%N Number of 3*n X 6 0..1 arrays with row sums 2 and column sums n.
%H Christoph Koutschan, <a href="/A172572/b172572.txt">Table of n, a(n) for n = 1..75</a> (first 33 terms from R. H. Hardin)
%H Manuel Kauers and Christoph Koutschan, <a href="https://arxiv.org/abs/2303.02793">Some D-finite and some possibly D-finite sequences in the OEIS</a>, arXiv:2303.02793 [cs.SC], 2023, pp. 19-23.
%F Conjectured recurrence of order 3 and degree 9: (n + 2)*(n + 3)^5*(2*n + 5)*(62*n^2 + 217*n + 191)*a(n + 3) - 9*(n + 2)*(3*n + 7)*(3*n + 8)*(5084*n^6 + 68634*n^5 + 383756*n^4 + 1137319*n^3 + 1884032*n^2 + 1653960*n + 601185)*a(n + 2) - 9*(3*n + 4)*(3*n + 5)*(3*n + 7)*(3*n + 8)*(31372*n^5 + 313720*n^4 + 1227805*n^3 + 2354425*n^2 + 2220988*n + 827860)*a(n + 1) + 20250*(n + 1)*(3*n + 1)*(3*n + 2)*(3*n + 4)*(3*n + 5)*(3*n + 7)*(3*n + 8)*(62*n^2 + 341*n + 470)*a(n) = 0. - _Christoph Koutschan_, Feb 26 2023
%F Conjecture: a(n) ~ 3^(3*n + 1/2) * 5^(3*n + 5/2) / (128 * Pi^(5/2) * n^(5/2)), based on the recurrence by _Christoph Koutschan_. - _Vaclav Kotesovec_, Feb 26 2023
%t Walks6D[n_Integer, steps_List] := Walks6D[n, steps] =
%t Module[{vals = {{{{{1}}}}}, seq = {}, n0, n1},
%t Do[
%t vals = Table[
%t n0 = Ceiling[(nsum - n1 - n2 - n3 - n4)/2];
%t Join[
%t Table[0, {n0}],
%t Table[Total[Function[s,
%t pos = Reverse[Sort[{n1, n2, n3, n4, n5, nsum - n1 - n2 - n3 - n4 - n5} - s]];
%t If[Min[pos] < 0, 0, vals[[##]]& @@ Most[pos + 1]]] /@ steps],
%t {n5, n0, Min[n4, nsum - n1 - n2 - n3 - n4]}]
%t ],
%t {n1, 0, Min[n, nsum]},
%t {n2, 0, Min[n1, nsum - n1]},
%t {n3, 0, Min[n2, nsum - n1 - n2]},
%t {n4, 0, Min[n3, nsum - n1 - n2 - n3]}];
%t If[IntegerQ[n1 = nsum/6 + 1], AppendTo[seq, vals[[n1, n1, n1, n1, n1]]]],
%t {nsum, 2, 6 n, 2}];
%t Return[seq];
%t ];
%t Walks6D[20, Permutations[{1, 1, 0, 0, 0, 0}]] (* _Christoph Koutschan_, Feb 26 2023 *)
%K nonn
%O 1,1
%A _R. H. Hardin_, Feb 06 2010
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