|
%I #9 Apr 03 2019 03:00:02
%S 1,1,9,39,155,570,2131,7599,26667,90996,305144,1004173,3254123,
%T 10385884,32704819,101678860,312435675,949498206,2855953018,
%U 8507079361,25108844890,73468004480,213201630328,613871526178,1754365814430,4978113020152,14029639217532,39281646364737
%N Expansion of Product_{k>=1} 1/(1 - x^k)^(k*binomial(k+2,3)).
%C Euler transform of A002417.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F G.f.: Product_{k>=1} 1/(1 - x^k)^A002417(k).
%F G.f.: exp(Sum_{k>=1} x^k*(1 + 3*x^k)/(k*(1 - x^k)^5)).
%F a(n) ~ 1/(2^(601/720) * 3^(359/480) * 7^(119/1440) * n^(839/1440) * Pi^(1/240)) * exp(-Zeta(3)/(12 * Pi^2) + (491 * Zeta(5))/(400 * Pi^4) - (2250423 * Zeta(5)^3)/(10 * Pi^14) + (103355177121 * Zeta(5)^5)/(10 * Pi^24) + Zeta'(-3)/2 + ((-7 * 7^(1/6) * Pi)/(1200 * 2^(1/3) * sqrt(3)) + (27783 * sqrt(3) * 7^(1/6) * Zeta(5)^2)/(40 * 2^(1/3) * Pi^9) - (614365479 * sqrt(3) * 7^(1/6) * Zeta(5)^4)/(16 * 2^(1/3) * Pi^19)) * n^(1/6) + ((-63 * 7^(1/3) * Zeta(5))/(10 * 2^(2/3) * Pi^4) + (214326 * 14^(1/3) * Zeta(5)^3)/Pi^14) * n^(1/3) + ((sqrt(7/3) * Pi)/30 - (1701 * sqrt(21) * Zeta(5)^2)/(2 * Pi^9)) * sqrt(n) + ((27 * 7^(2/3) * Zeta(5))/(2 * 2^(1/3) * Pi^4)) * n^(2/3) + ((2 * 2^(1/3) * sqrt(3) * Pi)/(5 * 7^(1/6))) * n^(5/6)). - _Vaclav Kotesovec_, Jul 28 2018
%p a:=series(mul(1/(1-x^k)^(k*binomial(k+2,3)),k=1..100),x=0,28): seq(coeff(a,x,n),n=0..27); # _Paolo P. Lava_, Apr 02 2019
%t nmax = 27; CoefficientList[Series[Product[1/(1 - x^k)^(k Binomial[k + 2, 3]), {k, 1, nmax}], {x, 0, nmax}], x]
%t nmax = 27; CoefficientList[Series[Exp[Sum[x^k (1 + 3 x^k)/(k (1 - x^k)^5), {k, 1, nmax}]], {x, 0, nmax}], x]
%t a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d^3 (d + 1) (d + 2)/6, {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 27}]
%Y Cf. A000391, A002417, A278767, A279217, A305653, A317017, A317020, A317021.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jul 19 2018
|
