OFFSET
0,3
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..55
FORMULA
G.f.: x/(1-x)^2 = Sum_{n>=0} a(n)*x^n/(1-4^n*x).
By using the inverse transformation: a(n) = Sum_{k=0..n} k*A118188(n-k)*4^(k*(n-k)) for n>=0.
a(2^n) is divisible by 2^n.
L.g.f.: Sum_{n>=1} a(n)*x^n/[n*2^(n^2)] = log( Sum_{n>=0} x^n/2^(n^2) ). - Paul D. Hanna, Oct 14 2009
EXAMPLE
Column 0 of log(A118185) = [0, 1, -2/2, 19/3, -764/4, 125701/5, ...].
The g.f. is illustrated by:
x/(1-x)^2 = x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 + ...
= x/(1-4*x) - 2*x^2/(1-16*x) + 19*x^3/(1-64*x) - 764*x^4/(1-256*x) + 125701*x^5/(1-1024*x) - 83499002*x^6/(1-4096*x) + 222705979399*x^7/(1-16384*x) + ...
From Paul D. Hanna, Oct 14 2009: (Start)
Illustrate the logarithmic g.f. by:
L(x) = x/2^1 - 2*x^2/(2*2^4) + 19*x^3/(3*2^9) - 764*x^4/(4*2^16) +- ...
where exp(L(x)) = 1 + x/2^1 + x^2/2^4 + x^3/2^9 + x^4/2^16 + ... (End)
MATHEMATICA
a[n_]{= a[n]= -Sum[4^(j*(n-j))*j*A118188[j], {j, 0, n}];
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Jun 29 2021 *)
PROG
(PARI) {a(n)=local(T=matrix(n+1, n+1, r, c, if(r>=c, (4^(c-1))^(r-c))), L=sum(m=1, #T, -(T^0-T)^m/m)); return(n*L[n+1, 1])}
(PARI) {a(n)=n*2^(n^2)*polcoeff(log(sum(m=0, n, x^m/2^(m^2))+x*O(x^n)), n)} \\ Paul D. Hanna, Oct 14 2009
(Sage)
@CachedFunction
def a(n): return (-1)*sum(4^(j*(n-j))*j*A118188(j) for j in (0..n))
[a(n) for n in (0..30)] # G. C. Greubel, Jun 29 2021
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Apr 15 2006
STATUS
approved