%I #7 Jan 30 2020 17:21:59

%S 1,1,1,2,2,1,5,7,3,1,16,28,15,4,1,64,127,85,26,5,1,308,650,531,192,40,

%T 6,1,1728,3737,3600,1551,365,57,7,1,11046,23996,26266,13416,3635,620,

%U 77,8,1,79065,170866,205353,122770,38556,7356,973,100,9,1

%N Triangle, read by rows, where the g.f. of column k, C_k(x), is defined by the recursion: C_k(x) = ( 1 + Sum_{n>=1} x^n*C_{n-1+k}(x) )^(k+1).

%C This is a variant of triangle A124328.

%H G. C. Greubel, <a href="/A127082/b127082.txt">Rows n = 0..50 of triangle, flattened</a>

%e C_k = [ 1 + x*C_k + x^2*C_{k+1} + x^3*C_{k+2} +... ]^(k+1).

%e The columns are generated by working backwards:

%e C_3 = [ 1 + x*C_3 + x^2*C_4 + x^3*C_5 + x^4*C_6 +... ]^4;

%e C_2 = [ 1 + x*C_2 + x^2*C_3 + x^3*C_4 + x^4*C_5 +... ]^3;

%e C_1 = [ 1 + x*C_1 + x^2*C_2 + x^3*C_3 + x^4*C_4 +... ]^2;

%e C_0 = [ 1 + x*C_0 + x^2*C_1 + x^3*C_2 + x^4*C_3 +... ]^1;

%e thus the row sums equal column 0 shift left.

%e The triangle begins:

%e 1;

%e 1, 1;

%e 2, 2, 1;

%e 5, 7, 3, 1;

%e 16, 28, 15, 4, 1;

%e 64, 127, 85, 26, 5, 1;

%e 308, 650, 531, 192, 40, 6, 1;

%e 1728, 3737, 3600, 1551, 365, 57, 7, 1;

%e 11046, 23996, 26266, 13416, 3635, 620, 77, 8, 1;

%e 79065, 170866, 205353, 122770, 38556, 7356, 973, 100, 9, 1;

%e 625049, 1338578, 1716582, 1180496, 429515, 92730, 13412, 1440, 126, 10, 1;

%t T[n_, k_]:= T[n, k]= If[k==n, 1, Coefficient[(1 + x*Sum[x^(r-k)*Sum[T[r, c], {c, k, r}], {r, k, n-1}] + x^(n+1))^(k+1), x, n-k]]; Table[T[n, k], {n,0,12}, {k, 0,n}]//Flatten (* _G. C. Greubel_, Jan 30 2020 *)

%o (PARI) {T(n,k)=if(n==k,1,polcoeff( (1 + x*sum(r=k,n-1,x^(r-k)*sum(c=k,r, T(r,c) ))+x*O(x^n))^(k+1),n-k))}

%Y Cf. variant: A124328;

%Y Columns: A127083, A127084, A127085, A127086, A127090 (central terms).

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Jan 04 2007