A124530 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*k) for n>=0, with R_0(y) = 1/(1-y).
1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 5, 1, 1, 1, 4, 10, 16, 1, 1, 1, 5, 16, 39, 62, 1, 1, 1, 6, 23, 71, 174, 274, 1, 1, 1, 7, 31, 113, 351, 858, 1332, 1, 1, 1, 8, 40, 166, 608, 1891, 4564, 6978, 1, 1, 1, 9, 50, 231, 961, 3535, 10888, 25793, 38873, 1, 1, 1, 10, 61, 309, 1427
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)^(2n) + y^3*R_3(y)^(3n) +... more explicitly: R_0 = 1 + y + y^2 + y^3 + y^4 + ... R_1 = 1 + y*(R_1)^1 + y^2*(R_2)^2 + y^3*(R_3)^3 + y^4*(R_4)^4 + ... R_2 = 1 + y*(R_1)^2 + y^2*(R_2)^4 + y^3*(R_3)^6 + y^4*(R_4)^8 +... R_3 = 1 + y*(R_1)^3 + y^2*(R_2)^6 + y^3*(R_3)^9 + y^4*(R_4)^12 +... R_4 = 1 + y*(R_1)^4 + y^2*(R_2)^8 + y^3*(R_3)^12 + y^4*(R_4)^16 +... etc., for all rows. Rectangular table begins: 1,1,1,1,1,1,1,1,1,1,1,1,1,... 1,1,2,5,16,62,274,1332,6978,38873,228090,1399625,8933506,... 1,1,3,10,39,174,858,4564,25793,153301,951325,6130757,40861658,... 1,1,4,16,71,351,1891,10888,66139,420235,2775981,18978873,... 1,1,5,23,113,608,3535,21844,141809,959882,6738850,48877221,... 1,1,6,31,166,961,5977,39363,271564,1949165,14487241,111115804,... 1,1,7,40,231,1427,9430,65810,480077,3637345,28502254,230271472,... 1,1,8,50,309,2024,14134,104028,798954,6363948,52370770,443997440,... 1,1,9,61,401,2771,20357,157383,1267833,10579140,91111871,... 1,1,10,73,508,3688,28396,229810,1935562,16866694,151563677,... 1,1,11,86,631,4796,38578,325860,2861457,25969694,242836861,... 1,1,12,100,771,6117,51261,450748,4116641,38819122,376841378,...
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)))))); R[n+1][k+1]
Formula
G.f.: A(x,y) = Sum_{n>=0} x^n*R_n(y) = Sum_{k>=0} y^k/(1 - x*R_k(y)^k).
Comments