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%I #3 Mar 30 2012 18:37:01
%S 1,1,1,1,3,1,1,8,6,1,1,22,28,11,1,1,65,120,81,20,1,1,209,500,494,219,
%T 37,1,1,730,2088,2733,1812,578,70,1,1,2743,8884,14411,12904,6299,1518,
%U 135,1,1,10958,38803,74484,84424,56590,21384,4007,264,1,1,46057,174366
%N Triangle, read by rows, where row n equals the inverse binomial transform of column n in the rectangular table A124460.
%C In table A124460, the g.f. of row n, R_n(y), simultaneously satisfies: R_n(y) = Sum_{k>=0} y^k * R_k(y)^n for n>=0.
%F Secondary diagonal T(n+1,n) = 2^n + n = A006127(n).
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 8, 6, 1;
%e 1, 22, 28, 11, 1;
%e 1, 65, 120, 81, 20, 1;
%e 1, 209, 500, 494, 219, 37, 1;
%e 1, 730, 2088, 2733, 1812, 578, 70, 1;
%e 1, 2743, 8884, 14411, 12904, 6299, 1518, 135, 1;
%e 1, 10958, 38803, 74484, 84424, 56590, 21384, 4007, 264, 1;
%e 1, 46057, 174366, 383391, 526121, 453082, 238853, 72076, 10693, 521, 1;
%o (PARI) {T(n,k)=local(R=vector(n+2,r,vector(n+2,c,binomial(r+c-2,c-1)))); for(i=0,n,for(r=0,n,R[r+1]=Vec(sum(c=0,n,x^c*Ser(R[c+1])^r+O(x^(n+1)))))); Vec(subst(Ser(vector(n+1,j,R[j][n+1])),x,x/(1+x))/(1+x))[k+1]}
%Y Cf. A124470 (row sums), A006127 (diagonal T(n+1, n)); A124460 (table).
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Nov 03 2006
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