OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} ((1+x)^n - 1)^n / A(x)^(n+1).
(2) 1 = Sum_{n>=0} (1+x)^(n^2) / (A(x) + (1+x)^n)^(n+1). - Paul D. Hanna, Dec 13 2018
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = 3.16108865386542881383... and c = 0.212154215724410476311... - Vaclav Kotesovec, Oct 06 2020
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 21*x^3 + 221*x^4 + 3117*x^5 + 54597*x^6 + 1136127*x^7 + 27293715*x^8 + 742143113*x^9 + 22512196673*x^10 + ...
such that
A(x) = 1 + ((1+x)-1)/A(x) + ((1+x)^2-1)^2/A(x)^2 + ((1+x)^3-1)^3/A(x)^3 + ((1+x)^4-1)^4/A(x)^4 + ((1+x)^5-1)^5/A(x)^5 + ((1+x)^6-1)^6/A(x)^6 + ...
also,
1 = 1/(A(x) + 1) + (1+x)/(A(x) + (1+x))^2 + (1+x)^4/(A(x) + (1+x)^2)^3 + (1+x)^9/(A(x) + (1+x)^3)^4 + (1+x)^16/(A(x) + (1+x)^4)^5 + (1+x)^25/(A(x) + (1+x)^5)^6 + (1+x)^36/(A(x) + (1+x)^6)^7 + ... + (1+x)^(n^2) / (A(x) + (1+x)^n)^(n+1) + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=1, n, A = concat(A, 0); A[#A] = Vec(sum(n=0, #A, ((1+x)^n - 1 +x*O(x^#A))^n / Ser(A)^(n+1) ) )[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 20 2018
STATUS
approved