A108990 Triangle, read by rows, where the g.f. of row n, R_n(x), is a polynomial of degree n that satisfies: [x^k] R_{n+1}(x) = [x^k] (1 + x*R_n(x))^(n+1) for k=0..n+1, with R_0(x) = 1.

1, 1, 1, 1, 2, 3, 1, 3, 9, 22, 1, 4, 18, 76, 287, 1, 5, 30, 180, 1025, 5496, 1, 6, 45, 350, 2670, 19656, 139727, 1, 7, 63, 602, 5761, 54096, 495964, 4447234, 1, 8, 84, 952, 10962, 125328, 1411228, 15629720, 170501039, 1, 9, 108, 1416, 19062, 257040

Offset

0,5

Links

Table of n, a(n) for n=0..50.

Example

The g.f. of row 4 is: 1 + 4*x + 18*x^2 + 76*x^3 + 287*x^4
and equals the initial coefficients from [x^0] to [x^4] in:
(1+ x*R_3(x))^4 = (1 + x*(1 + 3*x + 9*x^2 + 22*x^3))^4.
Triangle begins:
1;
1, 1;
1, 2, 3;
1, 3, 9, 22;
1, 4, 18, 76, 287;
1, 5, 30, 180, 1025, 5496;
1, 6, 45, 350, 2670, 19656, 139727;
1, 7, 63, 602, 5761, 54096, 495964, 4447234;
1, 8, 84, 952, 10962, 125328, 1411228, 15629720, 170501039;
1, 9, 108, 1416, 19062, 257040, 3437028, 45421776, 593136513, 7661336692;
1, 10, 135, 2010, 30975, 481392, 7459230, 114696840, 1747674405, 26392662280, 395314147319; ...

Prog

(PARI) T(n, k)=local(F=1+x*O(x^n)); if(n<k || k<0, 0, for(m=1, n, F=(1+x*F)^m); polcoeff(F, k))
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A108991, A108992, A108993, A108994, A108995, A108996.

Sequence in context: A259760 A010027 A151880 * A145080 A065078 A203989

Adjacent sequences: A108987 A108988 A108989 * A108991 A108992 A108993

Keyword

nonn,tabl

Author

Paul D. Hanna, Jun 15 2005

Status

approved