OFFSET
0,4
LINKS
G. C. Greubel, Rows n = 0..50 of triangle, flattened
EXAMPLE
C_k = [ 1 + x*C_{k+1} + x^2*C_{k+2} + x^3*C_{k+3} +... ]^(k+1).
The columns are generated by working backwards:
C_3 = [ 1 + x*C_4 + x^2*C_5 + x^3*C_6 + x^4*C_7 +... ]^4;
C_2 = [ 1 + x*C_3 + x^2*C_4 + x^3*C_5 + x^4*C_6 +... ]^3;
C_1 = [ 1 + x*C_2 + x^2*C_3 + x^3*C_4 + x^4*C_5 +... ]^2;
C_0 = [ 1 + x*C_1 + x^2*C_2 + x^3*C_3 + x^4*C_4 +... ]^1.
The triangle begins:
1;
1, 1;
3, 2, 1;
13, 9, 3, 1;
77, 54, 18, 4, 1;
587, 412, 139, 30, 5, 1;
5484, 3834, 1314, 284, 45, 6, 1;
60582, 42131, 14658, 3217, 505, 63, 7, 1;
771261, 533558, 188012, 42100, 6680, 818, 84, 8, 1;
11102828, 7645065, 2721462, 621936, 100621, 12387, 1239, 108, 9, 1; ...
MATHEMATICA
T[n_, k_]:= T[n, k]= If[k==n, 1, Coefficient[(1 + x*Sum[x^(r-k-1)*Sum[T[r, c], {c, k+1, r}], {r, k+1, n}] +x^(n+1))^(k+1), x, n-k]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 27 2020 *)
PROG
(PARI) {T(n, k) = if(n==k, 1, polcoeff( (1 + x*sum(r=k+1, n, x^(r-k-1)*sum(c=k+1, r, T(r, c))) +x*O(x^n))^(k+1), n-k))}
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jan 05 2007
STATUS
approved