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A124539 Triangle, read by rows, where row n equals the inverse binomial transform of column n in the rectangular table A124530.

%I A124539 #5 Jun 14 2017 00:12:29
%S A124539 1,1,0,1,1,0,1,4,1,0,1,15,8,1,0,1,61,51,14,1,0,1,273,311,138,24,1,0,1,
%T A124539 1331,1901,1191,349,42,1,0,1,6977,11838,9693,4100,868,76,1,0,1,38872,
%U A124539 75556,76950,43257,13459,2163,142,1,0,1,228089,495146,606275,430517
%N A124539 Triangle, read by rows, where row n equals the inverse binomial transform of column n in the rectangular table A124530.
%C A124539 In table A124530, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k*R_k(y)^(n*k) for n>=0.
%F A124539 Secondary diagonal T(n+1,n) = 2^n + 2n.
%e A124539 Triangle begins:
%e A124539 1;
%e A124539 1, 0;
%e A124539 1, 1, 0;
%e A124539 1, 4, 1, 0;
%e A124539 1, 15, 8, 1, 0;
%e A124539 1, 61, 51, 14, 1, 0;
%e A124539 1, 273, 311, 138, 24, 1, 0;
%e A124539 1, 1331, 1901, 1191, 349, 42, 1, 0;
%e A124539 1, 6977, 11838, 9693, 4100, 868, 76, 1, 0;
%e A124539 1, 38872, 75556, 76950, 43257, 13459, 2163, 142, 1, 0;
%e A124539 1, 228089, 495146, 606275, 430517, 180000, 43274, 5442, 272, 1, 0; ...
%o A124539 (PARI) T(n,k)=local(m=max(n,k),R);R=vector(m+1,r,vector(m+1,c,if(r==1 || c<=2,1,r^(c-2)))); for(i=0,m, for(r=0,m, R[r+1]=Vec(sum(c=0,m, x^c*Ser(R[c+1])^(r*c)+O(x^(m+1)))))); Vec(subst(Ser(vector(n+1, j, R[j][n+1])), x, x/(1+x))/(1+x))[k+1]
%Y A124539 Cf. A124530 (table).
%K A124539 nonn,tabl
%O A124539 0,8
%A A124539 _Paul D. Hanna_, Nov 05 2006