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 A209245 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}. 3
 1, 3, 33, 543, 10497, 220503, 4870401, 111243135, 2602452993, 61985744967, 1497148260033, 36566829737727, 901314269530113, 22385640256615743, 559574590912019457, 14065064484334380543, 355222860485671141377, 9008982166319523972903, 229325469394627488082497 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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. Slice x(1,j,k) with j,k >= 0 of the cube begins:   1,  1,  1,   1,   1,    1,    1,    1, ... A000012   1,  3,  5,   7,   9,   11,   13,   15, ... A005408   1,  5, 11,  19,  29,   41,   55,   71, ... A028387   1,  7, 19,  39,  69,  111,  167,  239, ... A108766(k+1)   1,  9, 29,  69, 139,  251,  419,  659, ...   1, 11, 41, 111, 251,  503,  923, 1583, ...   1, 13, 55, 167, 419,  923, 1847, 3431, ...   1, 15, 71, 239, 659, 1583, 3431, 6863, ... The main diagonal of the slice is A134760. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..300 FORMULA 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. a(n) ~ 3^(3*n+1/2) / (8*Pi*n). - Vaclav Kotesovec, Sep 07 2014 MAPLE a:= proc(n) option remember; `if`(n<2, 2*n+1,       ((888-3020*n+3668*n^2-1912*n^3+364*n^4) *a(n-1)        +3*(3*n-4)*(7*n-5)*(2*n-3)*(3*n-5) *a(n-2)) /        ((2*n-1)*(7*n-12)*(n-1)^2))     end: seq(a(n), n=0..20);  # Alois P. Heinz, Jan 17 2013 MATHEMATICA b[] = 0; b[args__] := b[args] = If[{args}[[1]] == 0, 1, Sum[b @@ Sort[ ReplacePart[{args}, i -> {args}[[i]] - 1]], {i, 1, Length[{args}]}]]; a[n_] := b @@ Table[n, 3]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 03 2018, from Alois P. Heinz's Maple code for A210472 *) CROSSREFS Cf. A164039, A134760, A209288. Column k=3 of A210472. - Alois P. Heinz, Jan 23 2013 Sequence in context: A336539 A221147 A291818 * A092170 A083080 A002916 Adjacent sequences:  A209242 A209243 A209244 * A209246 A209247 A209248 KEYWORD nonn AUTHOR Jon Perry, Jan 13 2013 STATUS approved

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Last modified August 14 10:47 EDT 2020. Contains 336480 sequences. (Running on oeis4.)