|
|
A023878
|
|
Expansion of Product_{k>=1} (1 - x^k)^(-k^9).
|
|
7
|
|
|
1, 1, 513, 20196, 413668, 12444489, 372960863, 9158023846, 223763768245, 5567490203192, 132000248840652, 3018181447183141, 68165389692659690, 1512302997486058542, 32793035921825542778, 698432551205542941608, 14654522099892985823429, 302753023792981375706399
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
In general, column m > 0 of A144048 is asymptotic to (Gamma(m+2)*Zeta(m+2))^((1-2*Zeta(-m))/(2*m+4)) * exp((m+2)/(m+1) * (Gamma(m+2)*Zeta(m+2))^(1/(m+2)) * n^((m+1)/(m+2)) + Zeta'(-m)) / (sqrt(2*Pi*(m+2)) * n^((m+3-2*Zeta(-m))/(2*m+4))). - Vaclav Kotesovec, Mar 01 2015
|
|
LINKS
|
|
|
FORMULA
|
a(n) ~ 3^(67/363) * 5^(67/726) * (7*Zeta(11))^(67/1452) * exp(11 * 3^(4/11) * n^(10/11) * (7*Zeta(11))^(1/11) / (2^(3/11) * 5^(9/11)) + Zeta'(-9)) / (2^(95/726) * sqrt(11*Pi) * n^(793/1452)), where Zeta(11) = A013669 = 1.00049418860411946..., Zeta'(-9) = (5*(7129/2520 - gamma - log(2*Pi))/66 + 14175*Zeta'(10) / (2*Pi^10))/10 = 0.00313014531978857275492576829... . - Vaclav Kotesovec, Feb 27 2015
a(n) = (1/n)*Sum_{k=1..n} sigma_10(k)*a(n-k). - Seiichi Manyama, Mar 05 2017
|
|
MAPLE
|
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1,
add(add(d*d^9, d=divisors(j)) *a(n-j), j=1..n)/n)
end:
|
|
MATHEMATICA
|
nmax=30; CoefficientList[Series[Product[1/(1-x^k)^(k^9), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 01 2015 *)
|
|
PROG
|
(PARI) m=30; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^9)) \\ G. C. Greubel, Oct 31 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^9: k in [1..m]]) )); // G. C. Greubel, Oct 3012018
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|