A305652 Expansion of Product_{k>=1} (1 + x^k)^(2^(k-1)-1).

1, 0, 1, 3, 7, 18, 41, 99, 227, 538, 1236, 2872, 6597, 15166, 34669, 79150, 180011, 408616, 925015, 2089607, 4709937, 10595275, 23788174, 53312366, 119271967, 266399612, 594077742, 1322815256, 2941225084, 6530659320, 14481362803, 32070677496, 70937233268, 156721128440

Offset

0,4

Comments

Weigh transform of A000225, shifted right one place.

Convolution of the sequences A081362 and A098407.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

N. J. A. Sloane, Transforms

Formula

G.f.: Product_{k>=1} (1 + x^k)^A000225(k-1).

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

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^(2*k)/(k*(1 - x^k)*(1 - 2*x^k))).

a(n) ~ 2^n * exp(sqrt(2*n) - 5/4 + c) / (sqrt(2*Pi) * 2^(3/4) * n^(3/4)), where c = Sum_{k>=2} -(-1)^k / (k*(2^k-1)*(2^k-2)) = -0.07640757130267274170429705262846... - Vaclav Kotesovec, Jun 08 2018

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(2^(i-1)-1, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Jun 07 2018

Mathematica

nmax = 33; CoefficientList[Series[Product[(1 + x^k)^(2^(k-1)-1), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 33; CoefficientList[Series[Exp[Sum[(-1)^(k + 1) x^(2 k)/(k (1 - x^k) (1 - 2 x^k)), {k, 1, nmax}]], {x, 0, nmax}], x]
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d (2^(d - 1) - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[a[n], {n, 0, 33}]

Crossrefs

Cf. A000225, A001906, A011782, A081362, A098407, A102866, A110045.

Sequence in context: A265007 A026533 A131630 * A356938 A208771 A036884

Adjacent sequences: A305649 A305650 A305651 * A305653 A305654 A305655

Keyword

nonn

Author

Ilya Gutkovskiy, Jun 07 2018

Status

approved