A126341 Numerators of the limit of coefficients of q in { [x^n] W(x,q) } when read backward from [q^(n*(n-1)/2)] to [q^(n*(n-1)/2 - (n-1))], where W satisfies: W(x,q) = exp( q*x*W(q*x,q) ).

1, 1, 1, 7, 2, 2, 85, 11, 65, 19, 357, 19, 111, 123, 81, 16891, 3631, 8167, 16033, 6011, 7537, 60563, 32179, 7273, 90269, 17117, 61879, 141653, 9545369, 450889, 3251089, 230189, 1743845, 2481389, 389671, 367333, 55945199, 733219, 40966169

Offset

0,4

Comments

When the fractions {A126341(k)/A126342(k), k>=1} are formatted as a triangle in which row n is then multiplied by n!, the result is integer triangle A126343.

Links

Table of n, a(n) for n=0..38.

Formula

A126341(n)/A126342(n) = A126265(n, n*(n-1)/2) / n! for n>=1.

Example

The function W that satisfies: W(x,q) = exp( q*x*W(q*x,q) ) begins:
W(x,q) = 1 + q*x + (1/2 + q)*q^2*x^2 +
(1/6 + 1*q + 1/2*q^2 + 1*q^3)*q^3*x^3 +
(1/24 + 1/2*q + 1*q^2 + 7/6*q^3 + 1*q^4 + 1/2*q^5 + 1*q^6)*q^4*x^4 +...
Coefficients of q in {[x^n] W(x,q)} tend to a limit when read backwards:
n=1: [1, 1/2];
n=2: [1, 1/2, 1, 1/6];
n=3: [1, 1/2, 1, 7/6, 1, 1/2, 1/24].
The limit of coefficients of q in { [x^n] W(x,q) } begins:
[1, 1/2, 1, 7/6, 2, 2, 85/24, 11/3, 65/12, 19/3, 357/40, 19/2, 111/8, 123/8, 81/4, 16891/720,...].

Prog

(PARI) {a(n)=local(W=1+x); for(i=0, n, W=exp(subst(x*W, x, q*x+O(x^(n+2))))); numerator(Vec(Vec(W)[n+2]+O(q^(n*(n+1)/2+2)))[n*(n-1)/2+1])}

Crossrefs

Cf. A126342 (denominators), A126343, A126265.

Sequence in context: A300304 A208647 A163981 * A324788 A354640 A078087

Adjacent sequences: A126338 A126339 A126340 * A126342 A126343 A126344

Keyword

frac,nonn

Author

Paul D. Hanna, Dec 25 2006

Status

approved