A126342 Denominators 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, 2, 1, 6, 1, 1, 24, 3, 12, 3, 40, 2, 8, 8, 4, 720, 120, 240, 360, 120, 120, 840, 360, 72, 720, 120, 360, 720, 40320, 1680, 10080, 630, 4032, 5040, 672, 560, 72576, 840, 40320, 120960, 1920, 40320, 24192, 8064, 2520, 3628800, 362880, 145152, 4536, 725760

Offset

0,2

Comments

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

Links

Table of n, a(n) for n=0..49.

Formula

A126341(n)/a(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))))); denominator(Vec(Vec(W)[n+2]+O(q^(n*(n+1)/2+2)))[n*(n-1)/2+1])}

Crossrefs

Cf. A126341 (numerators), A126343, A126265.

Sequence in context: A290318 A139547 A323855 * A345461 A229818 A324500

Adjacent sequences: A126339 A126340 A126341 * A126343 A126344 A126345

Keyword

frac,nonn,changed

Author

Paul D. Hanna, Dec 25 2006

Status

approved