A124540 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.
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, 40, 156, 369, 486, 274, 0, 1, 7, 57, 285, 939, 1959, 2398, 1332, 0, 1, 8, 77, 470, 1995, 5764, 10912, 12668, 6978, 0, 1, 9, 100, 721, 3756, 13976, 36248, 63543, 70863, 38873, 0
Offset: 0
Examples
Row g.f.s R_n(y) simultaneously satisfy: R_n(y) = [1 + y*R_1(y)^n + y^2*R_2(y)^n + y^3*R_3(y)^n +...]^n more explicitly: R_0 = [1 + y + y^2 + y^3 + y^4 + ...]^0 = 1; R_1 = [1 + y*(R_1)^1 + y^2*(R_2)^1 + y^3*(R_3)^1 + y^4*(R_4)^1 +...]^1; R_2 = [1 + y*(R_1)^2 + y^2*(R_2)^2 + y^3*(R_3)^2 + y^4*(R_4)^2 +...]^2; R_3 = [1 + y*(R_1)^3 + y^2*(R_2)^3 + y^3*(R_3)^3 + y^4*(R_4)^3 +...]^3; R_4 = [1 + y*(R_1)^4 + y^2*(R_2)^4 + y^3*(R_3)^4 + y^4*(R_4)^4 +...]^4; etc., for all rows. Table begins: 1,0,0,0,0,0,0,0,0,0,0,... 1,1,2,5,16,62,274,1332,6978,38873,228090,... 1,2,7,26,107,486,2398,12668,70863,416304,2552490,... 1,3,15,73,369,1959,10912,63543,385341,2424988,15788469,... 1,4,26,156,939,5764,36248,233900,1549193,10529052,73390856,... 1,5,40,285,1995,13976,98665,704810,5107950,37619020,281850156,... 1,6,57,470,3756,29658,233241,1836912,14543877,116087596,936035298,... 1,7,77,721,6482,57057,495922,4282895,36922550,318834341,2765000007,... 1,8,100,1048,10474,101800,970628,9140344,85445683,795971176,7410928800,... 1,9,126,1461,16074,171090,1777416,18151272,183201255,1834958107,... 1,10,155,1970,23665,273902,3081700,33954660,368443380,3954149640,... 1,11,187,2585,33671,421179,5104528,60398327,701775756,8042277034,... 1,12,222,3316,46557,626028,8133916,102916452,1275653922,15559229828,...
Crossrefs
Programs
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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]
Formula
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.
Comments