A120568 Triangle, read by rows, where the k-th column equals the k-th self-composition of column 1 (A120567) for k>=0, such that row sums equal column 1.

1, 0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 5, 6, 3, 1, 0, 15, 21, 12, 4, 1, 0, 53, 82, 54, 20, 5, 1, 0, 215, 351, 263, 110, 30, 6, 1, 0, 976, 1630, 1364, 644, 195, 42, 7, 1, 0, 4859, 8151, 7469, 3956, 1335, 315, 56, 8, 1, 0, 26150, 43627, 42971, 25304, 9505, 2470, 476, 72, 9, 1

Offset

1,8

Links

Paul D. Hanna, Rows n = 1..45, flattened.

Example

The successive iterations of the g.f. of column 1 begin:
F(x) = x + x^2 + 2*x^3 + 5*x^4 + 15*x^5 + 53*x^6 + 215*x^7 + 976*x^8+...
F(F(x)) = x + 2*x^2 + 6*x^3 + 21*x^4 + 82*x^5 + 351*x^6 + 1630*x^7 +...
F(F(F(x))) = x + 3*x^2 + 12*x^3 + 54*x^4 + 263*x^5 + 1364*x^6 +...
F(F(F(F(x)))) = x + 4*x^2 + 20*x^3 + 110*x^4 + 644*x^5 + 3956*x^6 +...
F(F(F(F(F(x))))) = x + 5*x^2 + 30*x^3 + 195*x^4 + 1335*x^5 +9505*x^6+...
Triangle begins:
1;
0, 1;
0, 1, 1;
0, 2, 2, 1;
0, 5, 6, 3, 1;
0, 15, 21, 12, 4, 1;
0, 53, 82, 54, 20, 5, 1;
0, 215, 351, 263, 110, 30, 6, 1;
0, 976, 1630, 1364, 644, 195, 42, 7, 1;
0, 4859, 8151, 7469, 3956, 1335, 315, 56, 8, 1;
0, 26150, 43627, 42971, 25304, 9505, 2470, 476, 72, 9, 1;
0, 150585, 248631, 258976, 167804, 69899, 20023, 4207, 684, 90, 10, 1;
where column k equals the k-th self-composition of column 1.

Prog

(PARI) {ITERATE(n, F, p=#F)=local(G=x); for(i=1, n, G=subst(F, x, G+x*O(x^p))); G}
{T(n, k)=local(F=x+x*O(x^n), G=F, H=x); for(i=1, n, for(j=1, n, G=subst(F, x, G); H=H+x^j*G); F=H; G=x+x*O(x^n); H=G); polcoeff(ITERATE(k, F), n-k)}
for(n=1, 15, for(k=0, n-1, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A120567 (column 1).

Sequence in context: A110281 A120059 A067347 * A321960 A189233 A242153

Adjacent sequences: A120565 A120566 A120567 * A120569 A120570 A120571

Keyword

nonn,tabl

Author

Paul D. Hanna, Jun 14 2006

Status

approved