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%I #32 Jun 03 2018 07:56:24
%S 1,4,196,22096,3323092,574346824,107697153304,21304602938056,
%T 4376897152490644,924871720044550888,199731547307306769736,
%U 43887077830441507774336,9780481173520567895278600,2205358814500087896152369104,502225405515985555630557626848
%N Main diagonal of the quadruple recurrence x(i,j,k,m) = x(i-1,j,k,m) + x(i,j-1,k,m) + x(i,j,k-1,m) + x(i,j,k,m-1) with x(i,j,k,m) = 1 if 0 in {i,j,k,m}.
%H Alois P. Heinz, <a href="/A209288/b209288.txt">Table of n, a(n) for n = 0..417</a>
%F a(n) = x(n,n,n,n) with x(i,j,k,m) = 1 if 0 in {i,j,k,m} and x(i,j,k,m) = x(i-1,j,k,m) + x(i,j-1,k,m) + x(i,j,k-1,m) + x(i,j,k,m-1) else.
%F a(n) ~ 2^(8*n-1/2) / (27*(Pi*n)^(3/2)). - _Vaclav Kotesovec_, Sep 07 2014
%F Recurrence: 2*(n-2)*(n-1)^3*(3*n - 4)*(3*n - 2)*(10773*n^5 - 127620*n^4 + 601635*n^3 - 1410376*n^2 + 1643420*n - 761136)*a(n) = (n-2)*(50320683*n^10 - 922567239*n^9 + 7517570148*n^8 - 35838081882*n^7 + 110640905811*n^6 - 231017836827*n^5 + 330199460678*n^4 - 318795408964*n^3 + 198794448664*n^2 - 72220580288*n + 11590694016)*a(n-1) - 2*(2*n - 3)*(43339779*n^10 - 841711662*n^9 + 7268645808*n^8 - 36726190830*n^7 + 120139923393*n^6 - 265623988980*n^5 + 401575152460*n^4 - 409434087632*n^3 + 269059885664*n^2 - 102737317696*n + 17273392896)*a(n-2) - 16*(2*n - 5)*(2*n - 3)*(3*n - 8)*(3*n - 7)*(4*n - 11)*(4*n - 9)*(10773*n^5 - 73755*n^4 + 198885*n^3 - 263461*n^2 + 170958*n - 43304)*a(n-3). - _Vaclav Kotesovec_, Sep 12 2016
%p b:= proc() option remember; `if`(args[1]=0, 1,
%p add(b(sort(subsop(i=args[i]-1, [args]))[]), i=1..nargs))
%p end:
%p a:= n-> b(n$4):
%p seq(a(n), n=0..15); # _Alois P. Heinz_, Jan 18 2013
%t b[] = 0; b[args__] := b[args] = If[{args}[[1]] == 0, 1, Sum[b @@ Sort[ ReplacePart[{args}, i -> {args}[[i]] - 1]], {i, 1, Length[{args}]}]];
%t a[n_] := b @@ Table[n, 4];
%t Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Jun 03 2018, after _Alois P. Heinz_ *)
%Y Cf. A209245.
%Y Column k=4 of A210472. - _Alois P. Heinz_, Jan 23 2013
%K nonn
%O 0,2
%A _Jon Perry_, Jan 16 2013
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