A304962 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(2^(k-1)).

1, 2, 6, 18, 50, 138, 374, 994, 2610, 6778, 17414, 44346, 112034, 280970, 700038, 1733706, 4269970, 10463154, 25518198, 61962458, 149839602, 360958306, 866405702, 2072579058, 4942074082, 11748730482, 27849974598, 65837539522, 155236876018, 365125130490, 856767548022

Offset: 0

Author

Ilya Gutkovskiy, May 22 2018

Keywords

• nonn

Comments

Convolution of the sequences A034691 and A098407.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Simon Plouffe, Conjectures of the OEIS, as of June 20, 2018.
• Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Master's thesis, 1992. [see page 24]
• N. J. A. Sloane, Transforms
• Index entries for sequences related to partitions
• Index entries for sequences related to compositions

Crossrefs

Cf. A011782, A015128, A034691, A098407, A156616, A261519, A302239.

Programs

Maple

g:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      2^(d-1), d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, i) option remember; `if`(n=0 or i=1, `if`(n>1, 0, 1),
      add(b(n-i*j, i-1)*binomial(2^(i-1), j), j=0..n/i))
    end:
a:= n-> add(g(n-j)*b(j$2), j=0..n):
seq(a(n), n=0..35);  # Alois P. Heinz, May 22 2018
# Maple program to compute c(n) from a(n) or a(n) from c(n).
with(numtheory):
andrews:=proc(liste) local n,z,serie,ls,i,d,aaa;
   n:=nops(liste);
aaa:=liste;
serie:=listtoseries(aaa,z,ogf):
ls:=series(ln(serie),z,n);
   [seq(coeff(ls,z,d),d=1..n)];
   [seq(elemmobius(%,i),i=1..n-1)]
end:
swerdna:=proc(liste) local n,i,z;
  n:=nops(liste);
  series(convert([seq((1-z^i)^(-liste[i]),i=1..n)],`*`),z,n);
  [seq(coeff(%,z,i),i=0..n-1)]
end:
elemmobius:=proc(liste,d) local k,rep;
   rep:=0;
   for k in divisors(d) do
      rep:=rep+liste[k]*mobius(iquo(d,k))/iquo(d,k)
   od;
   rep
end:
# Here andrews() finds the c(n) and swerdna() finds the a(n) if the c(n) are known.
# For ordinary partitions the c(n) are [1,1,1,1,1, ...].
# Simon Plouffe, Jun 20 2018

Mathematica

nmax = 30; CoefficientList[Series[Product[((1 + x^k)/(1 - x^k))^(2^(k - 1)), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} ((1 + x^k)/(1 - x^k))^A011782(k).

Euler transform of c(n) with g.f.: -x*(-2*x^2-x+2)/(-4*x^3+2*x^2+2*x-1). - Simon Plouffe, Jun 20 2018

a(n) ~ A247003 * 2^(n-1) * exp(2*sqrt(n) - 1/2 + c) / (sqrt(Pi)*n^(3/4)), where c = Sum_{k>=2} -(-1)^k / (k*(2^k-2)) = -0.207530918644117743551169251314627032... - Vaclav Kotesovec, Sep 15 2021