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A127082 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).

%I A127082 #7 Jan 30 2020 17:21:59
%S A127082 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 A127082 6,1,1728,3737,3600,1551,365,57,7,1,11046,23996,26266,13416,3635,620,
%U A127082 77,8,1,79065,170866,205353,122770,38556,7356,973,100,9,1
%N A127082 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 A127082 This is a variant of triangle A124328.
%H A127082 G. C. Greubel, <a href="/A127082/b127082.txt">Rows n = 0..50 of triangle, flattened</a>
%e A127082 C_k = [ 1 + x*C_k + x^2*C_{k+1} + x^3*C_{k+2} +... ]^(k+1).
%e A127082 The columns are generated by working backwards:
%e A127082 C_3 = [ 1 + x*C_3 + x^2*C_4 + x^3*C_5 + x^4*C_6 +... ]^4;
%e A127082 C_2 = [ 1 + x*C_2 + x^2*C_3 + x^3*C_4 + x^4*C_5 +... ]^3;
%e A127082 C_1 = [ 1 + x*C_1 + x^2*C_2 + x^3*C_3 + x^4*C_4 +... ]^2;
%e A127082 C_0 = [ 1 + x*C_0 + x^2*C_1 + x^3*C_2 + x^4*C_3 +... ]^1;
%e A127082 thus the row sums equal column 0 shift left.
%e A127082 The triangle begins:
%e A127082        1;
%e A127082        1,       1;
%e A127082        2,       2,       1;
%e A127082        5,       7,       3,       1;
%e A127082       16,      28,      15,       4,      1;
%e A127082       64,     127,      85,      26,      5,     1;
%e A127082      308,     650,     531,     192,     40,     6,     1;
%e A127082     1728,    3737,    3600,    1551,    365,    57,     7,    1;
%e A127082    11046,   23996,   26266,   13416,   3635,   620,    77,    8,   1;
%e A127082    79065,  170866,  205353,  122770,  38556,  7356,   973,  100,   9,  1;
%e A127082   625049, 1338578, 1716582, 1180496, 429515, 92730, 13412, 1440, 126, 10, 1;
%t A127082 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 A127082 (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 A127082 Cf. variant: A124328;
%Y A127082 Columns: A127083, A127084, A127085, A127086, A127090 (central terms).
%K A127082 nonn,tabl
%O A127082 0,4
%A A127082 _Paul D. Hanna_, Jan 04 2007