OFFSET
0,3
COMMENTS
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
G.f. A(x) satisfies:
(1) 1 = Sum_{n>=0} Product_{k=n..2*n-1} ( (1+x)^k - A(x) ).
(2) 1 = Sum_{n>=0} (1+x)^(n*(3*n-1)/2) / Product_{k=n..2*n} (1 + (1+x)^k*A(x)).
a(n) ~ c * d^n * n! / sqrt(n), where d = 4.6634423082484267335372948079179... and c = 0.10798828318774... - Vaclav Kotesovec, Oct 24 2020
EXAMPLE
G.f.: A(x) = 1 + x + 2*x^2 + 23*x^3 + 383*x^4 + 8456*x^5 + 228657*x^6 + 7268077*x^7 + 264627570*x^8 + 10842464809*x^9 + 493454895161*x^10 + ...
Let A = A(x), then g.f. A(x) satisfies
1 = 1 + ((1+x) - A) + ((1+x)^2 - A)*((1+x)^3 - A) + ((1+x)^3 - A)*((1+x)^4 - A)*((1+x)^5 - A) + ((1+x)^4 - A)*((1+x)^5 - A)*((1+x)^6 - A)*((1+x)^7 - A) + ((1+x)^5 - A)*((1+x)^6 - A)*((1+x)^7 - A)*((1+x)^8 - A)*((1+x)^9 - A) + ... + Product_{k=n..2*n-1} ((1+x)^k - A(x)) + ...
also
1 = 1/(1 + A) + (1+x)/((1 + (1+x)*A)*(1 + (1+x)^2*A)) + (1+x)^5/((1 + (1+x)^2*A)*(1 + (1+x)^3*A)*(1 + (1+x)^4*A)) + (1+x)^12/((1 + (1+x)^3*A)*(1 + (1+x)^4*A)*(1 + (1+x)^5*A)*(1 + (1+x)^6*A)) + (1+x)^22/((1 + (1+x)^4*A)*(1 + (1+x)^5*A)*(1 + (1+x)^6*A)*(1 + (1+x)^7*A)*(1 + (1+x)^8*A)) + ... + (1+x)^(n*(3*n-1)/2)/( Product_{k=n..2*n} (1 + (1+x)^(k*A(x)) ) + ...
PROG
(PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, prod(k=m, 2*m-1, (1+x)^k - Ser(A)) ) )[#A] ); A[n+1]}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 14 2020
STATUS
approved