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A304962 Expansion of Product_{k>=1} ((1 + x^k)/(1 - x^k))^(2^(k-1)). 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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 thesis 1992. [see page 24]

N. J. A. Sloane, Transforms

Index entries for sequences related to partitions

Index entries for sequences related to compositions

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

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]

CROSSREFS

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

Sequence in context: A048495 A089380 A271897 * A180282 A081154 A002900

Adjacent sequences:  A304959 A304960 A304961 * A304963 A304964 A304965

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, May 22 2018

STATUS

approved

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Last modified August 18 08:57 EDT 2019. Contains 326077 sequences. (Running on oeis4.)