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%I #28 Mar 10 2023 02:25:28
%S 90,202410,747558000,3536978063850,19292117692187340,
%T 115428185943399529200,737005538936597762145600,
%U 4937928427617947420104982250,34335031273255183438800013252500
%N Number of 3*n X 6 0..2 arrays with row sums 2 and column sums n.
%H Christoph Koutschan, <a href="/A172671/b172671.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 4 and degree 13: (n + 2)*(n + 3)^3*(3784*n^4 + 32164*n^3 + 100749*n^2 + 137862*n + 69678)*(n + 4)^5*a(n + 4) - (n + 2)*(n + 3)^3*(3*n + 10)*(3*n + 11)*(3799136*n^7 + 72183584*n^6 + 579689880*n^5 + 2548427912*n^4 + 6617561702*n^3 + 10141503096*n^2 + 8487349821*n + 2991586122)*a(n + 3) - 9*(n + 2)*(3*n + 7)*(3*n + 8)*(3*n + 10)*(3*n + 11)*(10844944*n^8 + 222321352*n^7 + 1973930222*n^6 + 9916013134*n^5 + 30831383530*n^4 + 60768378830*n^3 + 74160044251*n^2 + 51243135187*n + 15352797306)*a(n + 2) + 9*(3*n + 4)*(3*n + 5)*(3*n + 7)*(3*n + 8)*(3*n + 10)*(3*n + 11)*(29681696*n^7 + 504588832*n^6 + 3602458816*n^5 + 14001842392*n^4 + 32010306742*n^3 + 43078657918*n^2 + 31639900193*n + 9799573455)*a(n + 1) + 416745*(n + 1)*(3*n + 1)*(3*n + 2)*(3*n + 4)*(3*n + 5)*(3*n + 7)*(3*n + 8)*(3*n + 10)*(3*n + 11)*(3784*n^4 + 47300*n^3 + 219945*n^2 + 450988*n + 344237)*a(n) = 0. - _Christoph Koutschan_, Feb 26 2023
%F Conjecture: a(n) ~ 3^(3*n + 1/2) * 7^(3*n + 5/2) / (2^(19/2) * Pi^(5/2) * n^(5/2)), based on the recurrence by _Christoph Koutschan_. - _Vaclav Kotesovec_, Feb 27 2023
%p A172671:= proc(n) local x,i,j; coeftayl(add(add(x[i]*x[j],i=1..j),j=1..6)^(3*n),[seq(x[i],i=1..6)]=[0$6],[n$6]) end proc:
%p map(A172671, [$1..10]); # _Robert Israel_, Jan 15 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 rows = Join[Permutations[{1, 1, 0, 0, 0, 0}], Permutations[{2, 0, 0, 0, 0, 0}]];
%t Walks6D[20, rows] (* _Christoph Koutschan_, Feb 26 2023 *)
%K nonn
%O 1,1
%A _R. H. Hardin_, Feb 06 2010
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