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%I #6 Jun 14 2017 00:13:23
%S 1,1,0,1,1,0,1,2,2,0,1,3,7,5,0,1,4,15,26,16,0,1,5,26,73,107,62,0,1,6,
%T 40,156,369,486,274,0,1,7,57,285,939,1959,2398,1332,0,1,8,77,470,1995,
%U 5764,10912,12668,6978,0,1,9,100,721,3756,13976,36248,63543,70863,38873,0
%N Rectangular table, read by antidiagonals, such that the g.f. of row n, R_n(y), satisfies: R_n(y) = [ Sum_{k>=0} y^k * R_k(y)^n ]^n for n>=0, with R_0(y) = 1.
%C Antidiagonal sums equal row 1 (A124531).
%F Let S_n(y) be the g.f. of row n in table A124530, then R_n(y) = S_n(y)^n and so S_n(y) = Sum_{k>=0} y^k * R_k(y)^n for n>=0, where R_n(y) is the g.f. of row n in this table.
%e Row g.f.s R_n(y) simultaneously satisfy:
%e R_n(y) = [1 + y*R_1(y)^n + y^2*R_2(y)^n + y^3*R_3(y)^n +...]^n
%e more explicitly:
%e R_0 = [1 + y + y^2 + y^3 + y^4 + ...]^0 = 1;
%e R_1 = [1 + y*(R_1)^1 + y^2*(R_2)^1 + y^3*(R_3)^1 + y^4*(R_4)^1 +...]^1;
%e R_2 = [1 + y*(R_1)^2 + y^2*(R_2)^2 + y^3*(R_3)^2 + y^4*(R_4)^2 +...]^2;
%e R_3 = [1 + y*(R_1)^3 + y^2*(R_2)^3 + y^3*(R_3)^3 + y^4*(R_4)^3 +...]^3;
%e R_4 = [1 + y*(R_1)^4 + y^2*(R_2)^4 + y^3*(R_3)^4 + y^4*(R_4)^4 +...]^4;
%e etc., for all rows.
%e Table begins:
%e 1,0,0,0,0,0,0,0,0,0,0,...
%e 1,1,2,5,16,62,274,1332,6978,38873,228090,...
%e 1,2,7,26,107,486,2398,12668,70863,416304,2552490,...
%e 1,3,15,73,369,1959,10912,63543,385341,2424988,15788469,...
%e 1,4,26,156,939,5764,36248,233900,1549193,10529052,73390856,...
%e 1,5,40,285,1995,13976,98665,704810,5107950,37619020,281850156,...
%e 1,6,57,470,3756,29658,233241,1836912,14543877,116087596,936035298,...
%e 1,7,77,721,6482,57057,495922,4282895,36922550,318834341,2765000007,...
%e 1,8,100,1048,10474,101800,970628,9140344,85445683,795971176,7410928800,...
%e 1,9,126,1461,16074,171090,1777416,18151272,183201255,1834958107,...
%e 1,10,155,1970,23665,273902,3081700,33954660,368443380,3954149640,...
%e 1,11,187,2585,33671,421179,5104528,60398327,701775756,8042277034,...
%e 1,12,222,3316,46557,626028,8133916,102916452,1275653922,15559229828,...
%o (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(Ser(R[n+1])^n+O(x^(k+1)))[k+1]
%Y Rows: A124531, A124542, A124543, A124544, A124545, A124546; diagonals: A124547, A124548, A124549; related tables: A124530, A124550, A124460.
%K nonn,tabl
%O 0,8
%A _Paul D. Hanna_, Nov 05 2006