OFFSET
1,2
COMMENTS
Related identity: Sum_{n=-oo..+oo} x^n*(1 - x^n)^n = 0, which holds for |x| < 1.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..366
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) 1/x = Sum_{n=-oo..+oo} A(x)^n * (1 + A(x)^n)^(n+1).
(2) A(x) = x * Sum_{n=-oo..+oo} A(x)^(n^2) / (1 + A(x)^(n+1))^n.
(3) 0 = Sum_{n=-oo..+oo} (-1)^n * A(x)^n * (1 + A(x)^n)^n.
(4) 0 = Sum_{n=-oo..+oo} (-1)^n * A(x)^(n^2-n) / (1 + A(x)^n)^n.
EXAMPLE
G.f.: A(x) = x + 3*x^2 + 10*x^3 + 38*x^4 + 164*x^5 + 783*x^6 + 4005*x^7 + 21400*x^8 + 117602*x^9 + 659019*x^10 + 3748736*x^11 + 21588796*x^12 + ...
SPECIFIC VALUES.
A(t) = 1/3 at t = 0.14832728317680424382350400745104642263167027946862...
A(t) = 1/4 at t = 0.13433913917600443178696714330960568436967435856815...
A(t) = 1/5 at t = 0.12029812285398972879219940261295281978412524937754...
A(3/20) = 0.3521325903099608361455770617898033111722103407971...
A(1/7) = 0.29252723487814042698570516039406838227427731852655...
A(1/8) = 0.21500724214149512130643660913381998900575603076452...
A(1/9) = 0.17407688053908806913569913139334508111874650183559...
A(1/10) = 0.14711097488062849474543678333471254427936118296317...
PROG
(PARI) {a(n) = my(V=[0, 1], A); for(i=1, n, V=concat(V, 0); A = Ser(V);
V[#V] = polcoef( sum(m=-#A, #A, A^m*(1 + A^m)^(m+1) ), #V-3); ); polcoef(A, n)}
for(n=1, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn,new
AUTHOR
Paul D. Hanna, Dec 08 2024
STATUS
approved