

A126342


Denominators of the limit of coefficients of q in { [x^n] W(x,q) } when read backward from [q^(n*(n1)/2)] to [q^(n*(n1)/2  (n1))], where W satisfies: W(x,q) = exp( q*x*W(q*x,q) ).


3



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
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OFFSET

0,2


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..49.


FORMULA

A126341(n)/A126342(n) = A126265(n, n*(n1)/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*(n1)/2+1])}


CROSSREFS

Cf. A126341 (numerators), A126343, A126265.
Sequence in context: A290318 A139547 A323855 * A229818 A324500 A082388
Adjacent sequences: A126339 A126340 A126341 * A126343 A126344 A126345


KEYWORD

frac,nonn


AUTHOR

Paul D. Hanna, Dec 25 2006


STATUS

approved



