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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-Fran├žois 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: A243251 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 January 23 17:13 EST 2019. Contains 319399 sequences. (Running on oeis4.)