OFFSET
0,3
COMMENTS
Limit a(n)/a(n+1) = (sqrt(5) - 1)/2 = 0.6180339887... = A094214.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
FORMULA
G.f.: Sum_{n>=0} (1 + x^n*(1+x)^n)^n * x^n.
G.f.: Sum_{n>=0} (1+x)^(n^2) * x^(n*(n+1)) / (1 - x^(n+1)*(1+x)^n)^(n+1).
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 5*x^5 + 7*x^6 + 14*x^7 + 20*x^8 + 27*x^9 + 49*x^10 + 87*x^11 + 128*x^12 + 189*x^13 + 345*x^14 + 608*x^15 + ...
where
A(x) = 1 + (1 + x*(1+x))*x + (1 + x^2*(1+x)^2)^2*x^2 + (1 + x^3*(1+x)^3)^3*x^3 + (1 + x^4*(1+x)^4)^4*x^4 + ... + (1 + x^n*(1+x)^n)^n*x^n + ...
also
A(x) = 1/(1 - x) + (1+x)*x^2/(1 - x^2*(1+x))^2 + (1+x)^4*x^6/(1 - x^3*(1+x)^2)^3 + (1+x)^9*x^12/(1 - x^4*(1+x)^3)^4 + (1+x)^16*x^20/(1 - x^5*(1+x)^4)^5 + (1+x)^25*x^30/(1 - x^6*(1+x)^5)^6 + ... + (1+x)^(n^2)*x^(n*(n+1))/(1 - x^(n+1)*(1+x)^n)^(n+1) + ...
RELATED SERIES.
A(1/2) = Sum_{n>=0} (4^n + 3^n)^n / (2*4^n)^n = Sum_{n>=0} 2*4^n * 3^(n^2) / (2*4^n - 3^n)^(n+1) = 3.198779485217682672125325895125...
MATHEMATICA
nmax = 50; CoefficientList[Series[Sum[(1 + x^k*(1+x)^k)^k * x^k, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 18 2020 *)
PROG
(PARI) {a(n) = my(A=1); A = sum(m=0, n, (1 + x^m*(1+x)^m + x*O(x^n))^m * x^m); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))
(PARI) {a(n) = my(A=1); A = sum(m=0, sqrtint(n+1), (1+x + x*O(x^n))^(m^2) * x^(m*(m+1)) / (1 - x^(m+1)*(1+x)^m + x*O(x^n))^(m+1)); polcoeff(A, n)}
for(n=0, 50, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 17 2020
STATUS
approved