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%I #35 Jun 03 2018 02:08:55
%S 1,3,33,543,10497,220503,4870401,111243135,2602452993,61985744967,
%T 1497148260033,36566829737727,901314269530113,22385640256615743,
%U 559574590912019457,14065064484334380543,355222860485671141377,9008982166319523972903,229325469394627488082497
%N Main diagonal of the triple recurrence x(i,j,k) = x(i-1,j,k) + x(i,j-1,k) + x(i,j,k-1) with x(i,j,k) = 1 if 0 in {i,j,k}.
%C Level sums are defined as the sum of x(i,j,k) with i,j,k >= 0 and i+j+k = n. This gives 3*A164039(n-1) for n>0.
%C Slice x(1,j,k) with j,k >= 0 of the cube begins:
%C 1, 1, 1, 1, 1, 1, 1, 1, ... A000012
%C 1, 3, 5, 7, 9, 11, 13, 15, ... A005408
%C 1, 5, 11, 19, 29, 41, 55, 71, ... A028387
%C 1, 7, 19, 39, 69, 111, 167, 239, ... A108766(k+1)
%C 1, 9, 29, 69, 139, 251, 419, 659, ...
%C 1, 11, 41, 111, 251, 503, 923, 1583, ...
%C 1, 13, 55, 167, 419, 923, 1847, 3431, ...
%C 1, 15, 71, 239, 659, 1583, 3431, 6863, ...
%C The main diagonal of the slice is A134760.
%H Alois P. Heinz, <a href="/A209245/b209245.txt">Table of n, a(n) for n = 0..300</a>
%F a(n) = x(n,n,n) with x(i,j,k) = 1 if 0 in {i,j,k} and x(i,j,k) = x(i-1,j,k) + x(i,j-1,k) + x(i,j,k-1) else.
%F a(n) ~ 3^(3*n+1/2) / (8*Pi*n). - _Vaclav Kotesovec_, Sep 07 2014
%p a:= proc(n) option remember; `if`(n<2, 2*n+1,
%p ((888-3020*n+3668*n^2-1912*n^3+364*n^4) *a(n-1)
%p +3*(3*n-4)*(7*n-5)*(2*n-3)*(3*n-5) *a(n-2)) /
%p ((2*n-1)*(7*n-12)*(n-1)^2))
%p end:
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Jan 17 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, 3];
%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jun 03 2018, from _Alois P. Heinz_'s Maple code for A210472 *)
%Y Cf. A164039, A134760, A209288.
%Y Column k=3 of A210472. - _Alois P. Heinz_, Jan 23 2013
%K nonn
%O 0,2
%A _Jon Perry_, Jan 13 2013
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