A124530 - OEIS

%I #5 Jun 14 2017 00:11:21

%S 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,

%T 23,71,174,274,1,1,1,7,31,113,351,858,1332,1,1,1,8,40,166,608,1891,

%U 4564,6978,1,1,1,9,50,231,961,3535,10888,25793,38873,1,1,1,10,61,309,1427

%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*k) for n>=0, with R_0(y) = 1/(1-y).

%C See table A124540, in which row n equals the n-th self-convolution of row n of A124530 (this table).

%F G.f.: A(x,y) = Sum_{n>=0} x^n*R_n(y) = Sum_{k>=0} y^k/(1 - x*R_k(y)^k).

%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)^(2n) + y^3*R_3(y)^(3n) +...

%e more explicitly:

%e R_0 = 1 + y + y^2 + y^3 + y^4 + ...

%e R_1 = 1 + y*(R_1)^1 + y^2*(R_2)^2 + y^3*(R_3)^3 + y^4*(R_4)^4 + ...

%e R_2 = 1 + y*(R_1)^2 + y^2*(R_2)^4 + y^3*(R_3)^6 + y^4*(R_4)^8 +...

%e R_3 = 1 + y*(R_1)^3 + y^2*(R_2)^6 + y^3*(R_3)^9 + y^4*(R_4)^12 +...

%e R_4 = 1 + y*(R_1)^4 + y^2*(R_2)^8 + y^3*(R_3)^12 + y^4*(R_4)^16 +...

%e etc., for all rows.

%e Rectangular table begins:

%e 1,1,1,1,1,1,1,1,1,1,1,1,1,...

%e 1,1,2,5,16,62,274,1332,6978,38873,228090,1399625,8933506,...

%e 1,1,3,10,39,174,858,4564,25793,153301,951325,6130757,40861658,...

%e 1,1,4,16,71,351,1891,10888,66139,420235,2775981,18978873,...

%e 1,1,5,23,113,608,3535,21844,141809,959882,6738850,48877221,...

%e 1,1,6,31,166,961,5977,39363,271564,1949165,14487241,111115804,...

%e 1,1,7,40,231,1427,9430,65810,480077,3637345,28502254,230271472,...

%e 1,1,8,50,309,2024,14134,104028,798954,6363948,52370770,443997440,...

%e 1,1,9,61,401,2771,20357,157383,1267833,10579140,91111871,...

%e 1,1,10,73,508,3688,28396,229810,1935562,16866694,151563677,...

%e 1,1,11,86,631,4796,38578,325860,2861457,25969694,242836861,...

%e 1,1,12,100,771,6117,51261,450748,4116641,38819122,376841378,...

%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)))))); R[n+1][k+1]

%Y Rows: A124531, A124532, A124533, A124534, A124535, A124536; A124537 (diagonal), A124538 (antidiagonal sums); related tables: A124539, A124540; A124460 (variant).

%K nonn,tabl

%O 0,9

%A _Paul D. Hanna_, Nov 05 2006