OFFSET
0,2
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=3*n..4*n-1} ( (1+x)^k - A(x) ).
(2) 1 = Sum_{n>=0} (1+x)^(n*(7*n-1)/2) / Product_{k=3*n..4*n} (1 + (1+x)^k*A(x)).
a(n) ~ c * d^n * n^n / exp(n), where d = 11.0309918920494... and c = 0.314387643322... - Vaclav Kotesovec, Aug 12 2021
EXAMPLE
G.f.: A(x) = 1 + 3*x + 15*x^2 + 355*x^3 + 13876*x^4 + 722211*x^5 + 46171804*x^6 + 3473185910*x^7 + 299362847750*x^8 + 29037782693087*x^9 + ...
Let A = A(x), then g.f. A(x) satisfies
1 = 1 + ((1+x)^3 - A) + ((1+x)^6 - A)*((1+x)^7 - A) + ((1+x)^9 - A)*((1+x)^10 - A)*((1+x)^11 - A) + ((1+x)^12 - A)*((1+x)^13 - A)*((1+x)^14 - A)*((1+x)^15 - A) + ((1+x)^15 - A)*((1+x)^16 - A)*((1+x)^17 - A)*((1+x)^18 - A)*((1+x)^19 - A) + ... + Product_{k=3*n..4*n-1} ((1+x)^(n+k) - A(x)) + ...
also
1 = 1/(1 + A) + (1+x)^3/((1 + (1+x)^3*A)*(1 + (1+x)^4*A)) + (1+x)^13/((1 + (1+x)^6*A)*(1 + (1+x)^7*A)*(1 + (1+x)^8*A)) + (1+x)^30/((1 + (1+x)^9*A)*(1 + (1+x)^10*A)*(1 + (1+x)^11*A)*(1 + (1+x)^12*A)) + (1+x)^54/((1 + (1+x)^12*A)*(1 + (1+x)^13*A)*(1 + (1+x)^14*A)*(1 + (1+x)^15*A)*(1 + (1+x)^16*A)) + ... + (1+x)^(n*(7*n-1)/2)/( Product_{k=3*n..4*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=3*m, 4*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 19 2020
STATUS
approved