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

0,5

Comments

Antidiagonal sums forms row 1.

Links

Table of n, a(n) for n=0..69.

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.

Example

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,...

Prog

(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]}

Crossrefs

Rows: A124461, A124462, A124463, A124464, A124465, A124466; diagonals: A124467, A124468; related table: A124469.

Sequence in context: A089899 A092422 A096465 * A144042 A122084 A104559

Adjacent sequences: A124457 A124458 A124459 * A124461 A124462 A124463

Keyword

nonn,tabl

Author

Paul D. Hanna, Nov 03 2006

Status

approved