A124460 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).
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, 26, 73, 162, 251, 210, 1, 1, 8, 34, 111, 302, 637, 919, 731, 1, 1, 9, 43, 159, 506, 1325, 2622, 3549, 2744, 1, 1, 10, 53, 218, 788, 2437, 6032, 11188, 14371, 10959, 1, 1, 11, 64, 289
Offset: 0
Examples
Row o.g.f.s R_n(y) satisfy: R_n(y) = R_0(y)^n + y*R_1(y)^n + y^2*R_2(y)^n + y^3*R_3(y)^n +... more explicitly: R_0 = 1 + y + y^2 + y^3 + y^4 + ... R_1 = (R_0) + y*(R_1) + y^2*(R_2) + y^3*(R_3) + y^4*(R_4) + ... R_2 = (R_0)^2 + y*(R_1)^2 + y^2*(R_2)^2 + y^3*(R_3)^2 + y^4*(R_4)^2 +... R_3 = (R_0)^3 + y*(R_1)^3 + y^2*(R_2)^3 + y^3*(R_3)^3 + y^4*(R_4)^3 +... R_4 = (R_0)^4 + y*(R_1)^4 + y^2*(R_2)^4 + y^3*(R_3)^4 + y^4*(R_4)^4 +... etc., for all rows. Rectangular table begins: 1,1,1,1,1,1,1,1,1,1,1,1,... 1,2,4,9,23,66,210,731,2744,10959,46058,202028,... 1,3,8,23,73,251,919,3549,14371,60720,266481,1209807,... 1,4,13,44,162,637,2622,11188,49293,223768,1044661,5006126,... 1,5,19,73,302,1325,6032,28193,134825,659011,3290110,16764206,... 1,6,26,111,506,2437,12118,61499,317485,1666371,8891543,48221602,... 1,7,34,159,788,4117,22143,121079,670811,3764758,21408813,123367344,... 1,8,43,218,1163,6532,37703,220663,1304831,7795435,47075775,287431878,... 1,9,53,289,1647,9873,60767,378529,2377322,15055045,96196848,620412879,.. 1,10,64,373,2257,14356,93718,618367,4106995,27462836,185031258,... 1,11,76,471,3011,20223,139395,970217,6788744,47766886,338270681,... 1,12,89,584,3928,27743,201136,1471482,10811098,79794397,592228264,...
Crossrefs
Programs
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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]}
Formula
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.
Comments