A126342 - OEIS

%I #3 Mar 30 2012 18:37:02

%S 1,2,1,6,1,1,24,3,12,3,40,2,8,8,4,720,120,240,360,120,120,840,360,72,

%T 720,120,360,720,40320,1680,10080,630,4032,5040,672,560,72576,840,

%U 40320,120960,1920,40320,24192,8064,2520,3628800,362880,145152,4536,725760

%N 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) ).

%C 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.

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

%e The function W that satisfies: W(x,q) = exp( q*x*W(q*x,q) ) begins:

%e W(x,q) = 1 + q*x + (1/2 + q)*q^2*x^2 +

%e (1/6 + 1*q + 1/2*q^2 + 1*q^3)*q^3*x^3 +

%e (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 +...

%e Coefficients of q in {[x^n] W(x,q)} tend to a limit when read backwards:

%e n=1: [1, 1/2];

%e n=2: [1, 1/2, 1, 1/6];

%e n=3: [1, 1/2, 1, 7/6, 1, 1/2, 1/24].

%e The limit of coefficients of q in { [x^n] W(x,q) } begins:

%e [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,...].

%o (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])}

%Y Cf. A126341 (numerators), A126343, A126265.

%K frac,nonn

%O 0,2

%A _Paul D. Hanna_, Dec 25 2006