A127126 Triangle, read by rows, where the g.f. of column k, C_k(x), is defined by the recurrence: C_k(x) = [ 1 + Sum_{n>=k+1} C_n(x)*x^(n-k) ]^(k+1) for k>=0.

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

Offset: 0

Author

Paul D. Hanna, Jan 05 2007

Keywords

• nonn
• tabl

Comments

This is a variant of triangles: A127082, A124328.

Examples

			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; ...
		

Links

• G. C. Greubel, Rows n = 0..50 of triangle, flattened

Crossrefs

Columns: A127127, A127128, A127129, A127130.

Central terms: A127134.

Variants: A127082, A124328.

Programs

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 *)

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))}