OFFSET
0,3
COMMENTS
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..300
FORMULA
G.f.: Sum_{n>=1} (1-x)^n * (1 - (1-x)^n)^(n-1).
a(n) = c * n! / (sqrt(n) * (log(2))^(2*n)), where c = 0.93418651575946259471737... . - Vaclav Kotesovec, May 06 2014
In closed form, c = 2^(log(2)/2-1) / (log(2) * sqrt(Pi*(1-log(2)))). - Vaclav Kotesovec, May 03 2015
EXAMPLE
G.f.: A(x) = 1 + x + 4*x^2 + 23*x^3 + 176*x^4 + 1697*x^5 + 19805*x^6 +...
where the g.f. satisfies the identities:
(1) A(x) = 1 + x + (2*x - x^2)^2 + (3*x - 3*x^2 + x^3)^3 + (4*x - 6*x^2 + 4*x^3 - x^4)^4 + (5*x - 10*x^2 + 10*x^3 - 5*x^4 + x^5)^5 +...
(2) A(x) = (1-x) + (1-x)^2*(2*x - x^2) + (1-x)^3*(3*x - 3*x^2 + x^3)^2 + (1-x)^4*(4*x - 6*x^2 + 4*x^3 - x^4)^3 + (1-x)^5*(5*x - 10*x^2 + 10*x^3 - 5*x^4 + x^5)^4 +...
MATHEMATICA
terms = 24;
gf = 1 + Sum[(1 - (1 - x)^n)^n, {n, 1, terms}] + O[x]^terms;
CoefficientList[gf, x] (* Jean-François Alcover, Jul 01 2018 *)
PROG
(PARI) {a(n)=local(q=1/(1-x+x*O(x^n)), A=1); A=sum(k=0, n, q^(-k^2)*(q^k-1)^k); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n)=local(q=1/(1-x+x*O(x^n)), A=1); A=sum(k=1, n+1, q^(-k^2)*(q^k-1)^(k-1)); polcoeff(A, n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Dec 11 2012
EXTENSIONS
a(22)-a(23) corrected by Andrew Howroyd, Feb 22 2018
STATUS
approved