_Roger L. Bagula _ and _Gary W. Adamson (rlbagulatftn(AT)yahoo.com), _, Mar 28 2009

Triangle T(n,k) =3^(k-1)*e(n,k) read by rows, where e(n,k)= (e(n - 1, k)*e(n, k - 1) + 1)/e(n - 1, k - 1).

3, 5, 1, 7, 2, 27, 9, 3, 54, 9, 11, 4, 81, 18, 243, 13, 5, 108, 27, 486, 81, 15, 6, 135, 36, 729, 162, 2187, 17, 7, 162, 45, 972, 243, 4374, 729, 19, 8, 189, 54, 1215, 324, 6561, 1458, 19683, 21, 9, 216, 63, 1458, 405, 8748, 2187, 39366, 6561

1,1

H. S. M. Coxeter, Regular Polytopes, 3rd ed., Dover, NY, 1973, pp 159-162.

Row sums are (5-(-1)^n)*3^n/4-3*n/2.

T(n,k) = 3^(k-1)*e(n,k) where e(n,k)= ( 1+e(n-1,k)*e(n,k-1) )/e(n-1,k-1) and e(n,1)=2*n+1 define a triangle of fractions.

{3},

{5, 1},

{7, 2, 27},

{9, 3, 54, 9},

{11, 4, 81, 18, 243},

{13, 5, 108, 27, 486, 81},

{15, 6, 135, 36, 729, 162, 2187},

{17, 7, 162, 45, 972, 243, 4374, 729},

{19, 8, 189, 54, 1215, 324, 6561, 1458, 19683},

{21, 9, 216, 63, 1458, 405, 8748, 2187, 39366, 6561}

Clear[e, n, k];

e[n_, 0] := 2*n + 1;

e[n_, k_] := 0 /; k >= n;

e[n_, k_] := (e[n - 1, k]*e[n, k - 1] + 1)/e[n - 1, k - 1];

Table[Table[3^k*e[n, k], {k, 0, n - 1}], {n, 1, 10}];

Flatten[%]

A130303

nonn,tabl

Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Mar 28 2009

Edited by the Associate Editors of the OEIS, Apr 22 2009

approved