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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.

%I A127126 #10 Jan 28 2020 01:05:23
%S A127126 1,1,1,3,2,1,13,9,3,1,77,54,18,4,1,587,412,139,30,5,1,5484,3834,1314,
%T A127126 284,45,6,1,60582,42131,14658,3217,505,63,7,1,771261,533558,188012,
%U A127126 42100,6680,818,84,8,1,11102828,7645065,2721462,621936,100621,12387,1239,108,9,1
%N 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.
%C A127126 This is a variant of triangles: A127082, A124328.
%H A127126 G. C. Greubel, <a href="/A127126/b127126.txt">Rows n = 0..50 of triangle, flattened</a>
%e A127126 C_k = [ 1 + x*C_{k+1} + x^2*C_{k+2} + x^3*C_{k+3} +... ]^(k+1).
%e A127126 The columns are generated by working backwards:
%e A127126 C_3 = [ 1 + x*C_4 + x^2*C_5 + x^3*C_6 + x^4*C_7 +... ]^4;
%e A127126 C_2 = [ 1 + x*C_3 + x^2*C_4 + x^3*C_5 + x^4*C_6 +... ]^3;
%e A127126 C_1 = [ 1 + x*C_2 + x^2*C_3 + x^3*C_4 + x^4*C_5 +... ]^2;
%e A127126 C_0 = [ 1 + x*C_1 + x^2*C_2 + x^3*C_3 + x^4*C_4 +... ]^1.
%e A127126 The triangle begins:
%e A127126          1;
%e A127126          1,       1;
%e A127126          3,       2,       1;
%e A127126         13,       9,       3,      1;
%e A127126         77,      54,      18,      4,      1;
%e A127126        587,     412,     139,     30,      5,     1;
%e A127126       5484,    3834,    1314,    284,     45,     6,    1;
%e A127126      60582,   42131,   14658,   3217,    505,    63,    7,   1;
%e A127126     771261,  533558,  188012,  42100,   6680,   818,   84,   8, 1;
%e A127126   11102828, 7645065, 2721462, 621936, 100621, 12387, 1239, 108, 9, 1; ...
%t A127126 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 *)
%o A127126 (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))}
%Y A127126 Columns: A127127, A127128, A127129, A127130.
%Y A127126 Central terms: A127134.
%Y A127126 Variants: A127082, A124328.
%K A127126 nonn,tabl
%O A127126 0,4
%A A127126 _Paul D. Hanna_, Jan 05 2007