OFFSET
0,2
COMMENTS
In general, if g.f. = product_{j>=1} 1/(1-x^j)^binomial(j+k-1,k-1), k>=1, then log(a(n)) ~ (1+1/k) * k^(1/(k+1)) * Zeta(k+1)^(1/(k+1)) * n^(k/(k+1)).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, Graph - The asymptotic ratio
FORMULA
G.f.: Product_{j>=1} 1/(1-x^j)^C(j+4,4).
a(n) ~ Pi^(49/288) * exp(25/144 - 105*Zeta(3) / (8*Pi^2) + 5*Zeta'(-3)/12 + 29299*Zeta(5) / (128*Pi^4) + 2480625 * Zeta(3) * Zeta(5)^2 / (2*Pi^12) - 72930375 * Zeta(5)^3 / (2*Pi^14) + 1063324867500 * Zeta(5)^5 / Pi^24 + 41 * 7^(1/6) * Pi * n^(1/6) / (768*3^(1/2)) - 2625 * 3^(1/2) * 7^(1/6) * Zeta(3) * Zeta(5) * n^(1/6) / (2*Pi^7) + 540225 * 3^(1/2) * 7^(1/6) * Zeta(5)^2 * n^(1/6) / (16*Pi^9) - 4740474375 * 3^(1/2) * 7^(1/6) * Zeta(5)^4 * n^(1/6) / (4*Pi^19) + 25 * 7^(1/3) * Zeta(3) * n^(1/3) / (4*Pi^2) - 735 * 7^(1/3) * Zeta(5) * n^(1/3) / (8*Pi^4) + 3969000 * 7^(1/3) * Zeta(5)^3 * n^(1/3) / Pi^14 + 7^(3/2) * Pi * n^(1/2) / (3^(3/2)*8) - 4725 * 21^(1/2) * Zeta(5)^2 * n^(1/2) / Pi^9 + 45 * 7^(2/3) * Zeta(5) * n^(2/3) / (2*Pi^4) + 2 * 3^(1/2) * Pi * n^(5/6) / (5 * 7^(1/6))) / (A^(25/12) * 2^(3/2) * 3^(625/576) * 7^(337/1728) * n^(1201/1728)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant, Zeta(3) = A002117 = 1.202056903..., Zeta(5) = A013663 = 1.036927755... and Zeta'(-3) = ((gamma + log(2*Pi) - 11/6)/30 - 3*Zeta'(4)/Pi^4)/4 = 0.0053785763577743... .
MAPLE
with(numtheory):
a:= proc(n) option remember; local d, j; `if`(n=0, 1,
add(add(d*binomial(d+4, 4), d=divisors(j))*a(n-j), j=1..n)/n)
end:
seq(a(n), n=0..50); # after Alois P. Heinz
MATHEMATICA
nmax=50; CoefficientList[Series[Product[1/(1-x^j)^Binomial[j+4, 4], {j, 1, nmax}], {x, 0, nmax}], x]
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Mar 08 2015
STATUS
approved