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%I #3 Mar 30 2012 18:37:01
%S 1,1,1,1,2,1,1,3,4,1,1,4,8,9,1,1,5,13,23,23,1,1,6,19,44,73,66,1,1,7,
%T 26,73,162,251,210,1,1,8,34,111,302,637,919,731,1,1,9,43,159,506,1325,
%U 2622,3549,2744,1,1,10,53,218,788,2437,6032,11188,14371,10959,1,1,11,64,289
%N Rectangular table, read by antidiagonals, such that the o.g.f. of row n, R_n(y), satisfies: R_n(y) = Sum_{k>=0} y^k * R_k(y)^n for n>=0, with R_0(y) = 1/(1-y).
%C Antidiagonal sums forms row 1.
%F O.g.f.: A(x,y) = Sum_{n>=0} x^n*R_n(y) = Sum_{k>=0} y^k/(1 - x*R_k(y)) ; E.g.f.: A(x,y) = Sum_{n>=0} x^n*R_n(y)/n! = Sum_{k>=0} y^k*exp(x*R_k(y)) where R_n(y) is the o.g.f. of row n.
%e Row o.g.f.s R_n(y) satisfy:
%e R_n(y) = R_0(y)^n + y*R_1(y)^n + y^2*R_2(y)^n + y^3*R_3(y)^n +...
%e more explicitly:
%e R_0 = 1 + y + y^2 + y^3 + y^4 + ...
%e R_1 = (R_0) + y*(R_1) + y^2*(R_2) + y^3*(R_3) + y^4*(R_4) + ...
%e R_2 = (R_0)^2 + y*(R_1)^2 + y^2*(R_2)^2 + y^3*(R_3)^2 + y^4*(R_4)^2 +...
%e R_3 = (R_0)^3 + y*(R_1)^3 + y^2*(R_2)^3 + y^3*(R_3)^3 + y^4*(R_4)^3 +...
%e R_4 = (R_0)^4 + y*(R_1)^4 + y^2*(R_2)^4 + y^3*(R_3)^4 + y^4*(R_4)^4 +...
%e etc., for all rows.
%e Rectangular table begins:
%e 1,1,1,1,1,1,1,1,1,1,1,1,...
%e 1,2,4,9,23,66,210,731,2744,10959,46058,202028,...
%e 1,3,8,23,73,251,919,3549,14371,60720,266481,1209807,...
%e 1,4,13,44,162,637,2622,11188,49293,223768,1044661,5006126,...
%e 1,5,19,73,302,1325,6032,28193,134825,659011,3290110,16764206,...
%e 1,6,26,111,506,2437,12118,61499,317485,1666371,8891543,48221602,...
%e 1,7,34,159,788,4117,22143,121079,670811,3764758,21408813,123367344,...
%e 1,8,43,218,1163,6532,37703,220663,1304831,7795435,47075775,287431878,...
%e 1,9,53,289,1647,9873,60767,378529,2377322,15055045,96196848,620412879,..
%e 1,10,64,373,2257,14356,93718,618367,4106995,27462836,185031258,...
%e 1,11,76,471,3011,20223,139395,970217,6788744,47766886,338270681,...
%e 1,12,89,584,3928,27743,201136,1471482,10811098,79794397,592228264,...
%o (PARI) {T(n,k)=local(m=max(n,k),R=vector(m+1,r,vector(m+1,c,binomial(r+c-2,c-1)))); for(i=0,m,for(r=0,m,R[r+1]=Vec(sum(c=0,m,x^c*Ser(R[c+1])^r+O(x^(m+1))))));R[n+1][k+1]}
%Y Rows: A124461, A124462, A124463, A124464, A124465, A124466; diagonals: A124467, A124468; related table: A124469.
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Nov 03 2006
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