OFFSET
0,2
COMMENTS
In general, for m >= 0, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(k^m), then a(n) ~ ((2^(m+2)-1) * Gamma(m+2) * Zeta(m+2) / (2^(2*m+3) * n))^((1-2*Zeta(-m))/(2*m+4)) * exp((m+2)/(m+1) * ((2^(m+2)-1) * n^(m+1) * Gamma(m+2) * Zeta(m+2) / 2^(m+1))^(1/(m+2)) + Zeta'(-m)) / sqrt((m+2)*Pi*n). - Vaclav Kotesovec, Aug 19 2015
If m is even and m >= 2, then can be simplified as: a(n) ~ ((2^(m+2)-1) * Gamma(m+2) * Zeta(m+2) / (2^(2*m+3) * n))^(1/(2*m+4)) * exp((m+2)/(m+1) * ((2^(m+2)-1) * n^(m+1) * Gamma(m+2) * Zeta(m+2) / 2^(m+1))^(1/(m+2)) + (-1)^(m/2) * Gamma(m+1) * Zeta(m+1) / (2^(m+1) * Pi^m)) / sqrt((m+2)*Pi*n). - Vaclav Kotesovec, Aug 19 2015
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..2916 (terms 0..1000 from Vaclav Kotesovec)
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 23.
FORMULA
G.f.: exp( Sum_{n>=1} (sigma_5(2*n) - sigma_5(n))/16 * x^n/n ), where sigma_5(n) is the sum of 5th powers of divisors of n (A001160).
Inverse Euler transform has g.f.: x*(2 + 31*x + 152*x^2 + 341*x^3 + 460*x^4 + 341*x^5 + 152*x^6 + 31*x^7 + 2*x^8)/(1-x^2)^5.
a(n) ~ exp(3*2^(2/3)*Pi*n^(5/6)/5 + 3*Zeta(5)/(4*Pi^4)) / (2^(7/6) * 3^(1/2) * n^(7/12)), where Zeta(5) = A013663. - Vaclav Kotesovec, Aug 19 2015
a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A096960(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 30 2017
EXAMPLE
G.f.: A(x) = 1 + 2*x + 18*x^2 + 88*x^3 + 398*x^4 + 1768*x^5 + 7508*x^6 +...
where A(x) = (1+x)/(1-x) * (1+x^2)^16/(1-x^2)^16 * (1+x^3)^81/(1-x^3)^81 *...
Also, A(x) = Euler transform of [2,31,162,496,1250,2511,4802,7936,...]:
A(x) = 1/((1-x)^2*(1-x^2)^31*(1-x^3)^162*(1-x^4)^496*(1-x^5)^1250*(1-x^6)^2511*...).
MATHEMATICA
nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^(k^4), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2015 *)
PROG
(PARI) {a(n)=polcoeff(prod(m=1, n+1, ((1+x^m)/(1-x^m+x*O(x^n)))^(m^4)), n)}
(PARI) {a(n)=polcoeff(exp(sum(m=1, n, (sigma(2*m, 5)-sigma(m, 5))/16*x^m/m)+x*O(x^n)), n)}
(PARI) {a(n)=local(InvEulerGF=x*(2+31*x+152*x^2+341*x^3+460*x^4+341*x^5+152*x^6+31*x^7+2*x^8)/(1-x^2+x*O(x^n))^5); polcoeff(1/prod(k=1, n, (1-x^k+x*O(x^n))^polcoeff(InvEulerGF, k)), n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 12 2012
STATUS
approved