A261562 - OEIS

%I #22 Sep 21 2018 13:43:07

%S 1,2,4,14,24,58,124,238,480,922,1764,3238,6008,10794,19292,34166,

%T 59504,103042,176452,299958,505240,845570,1403324,2315118,3794640,

%U 6180370,10009540,16121374,25829512,41171690,65320956,103140062,162149488,253823178,395698276

%N Expansion of Product_{k>=1} (1 + 2*x^k)^k.

%H Alois P. Heinz, <a href="/A261562/b261562.txt">Table of n, a(n) for n = 0..5000</a> (first 2001 terms from Vaclav Kotesovec)

%F G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} -(-2)^d * n^2/d^2 ). - _Paul D. Hanna_, Sep 30 2015

%F a(n) ~ c^(1/6) * exp(3^(2/3)*c^(1/3)*n^(2/3)/2) / (3^(3/4)*sqrt(2*Pi)*n^(2/3)), where c = Pi^2*log(2) + log(2)^3 - 6*polylog(3, -1/2) = 10.00970018379942727227807189532511265744588249928680712064... . - _Vaclav Kotesovec_, Jan 04 2016

%p b:= proc(n, i) option remember;  `if`(n=0, 1, `if`(i<1, 0,

%p       add(2^j*binomial(i, j)*b(n-i*j, i-1), j=0..n/i)))

%p     end:

%p a:= n-> b(n$2):

%p seq(a(n), n=0..40);  # _Alois P. Heinz_, Sep 21 2018

%t nmax = 50; CoefficientList[Series[Product[(1 + 2*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]

%t nmax = 50; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*2^k/k*x^k/(1 - x^k)^2, {k, 1, nmax}]], {x, 0, nmax}], x]

%t nmax = 50; s = 1+2*x; Do[s*=Sum[Binomial[k, j]*2^j*x^(j*k), {j, 0, nmax/k}]; s = Take[Expand[s], Min[nmax + 1, Exponent[s, x] + 1]];, {k, 2, nmax}]; CoefficientList[s, x] (* _Vaclav Kotesovec_, Jan 08 2016 *)

%o (PARI) {a(n) = polcoeff( exp( sum(m=1, n, x^m/m * sumdiv(m, d, -(-2)^d * m^2/d^2) ) +x*O(x^n)), n)}

%o for(n=0, 40, print1(a(n), ", ")) \\ _Paul D. Hanna_, Sep 30 2015

%Y Cf. A026007, A032302, A261561, A261563, A266857, A266891.

%K nonn

%O 0,2

%A _Vaclav Kotesovec_, Aug 24 2015