A307058 Expansion of 1/(2 - Product_{k>=1} (1 + x^(2*k-1))).

1, 1, 1, 2, 4, 7, 12, 21, 38, 68, 120, 212, 377, 670, 1188, 2107, 3740, 6638, 11778, 20898, 37084, 65808, 116775, 207212, 367696, 652478, 1157815, 2054524, 3645730, 6469316, 11479734, 20370656, 36147506, 64143372, 113821732, 201975429, 358403220, 635982680, 1128544452, 2002589998

Offset: 0

Author

Ilya Gutkovskiy, Mar 21 2019

Keywords

• nonn

Comments

Invert transform of A000700.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000

Crossrefs

Cf. A000700, A055887, A302017, A304969, A307059.

Row sums of A341279.

Programs

Magma

m:=80;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( 1/(2 - (&*[1 + x^(2*j-1): j in [1..m+2]])) )); // G. C. Greubel, Jan 24 2024

Maple

g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(a(n-i)*g(i), i=1..n))
    end:
seq(a(n), n=0..39);  # Alois P. Heinz, Feb 09 2021

Mathematica

nmax = 39; CoefficientList[Series[1/(2 - Product[(1 + x^(2 k - 1)), {k, 1, nmax}]), {x, 0, nmax}], x]

SageMath

m=80;
def f(x): return 1/(2 - product(1+x^(2*j-1) for j in range(1,m+3)))
def A307058_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307058_list(m) # G. C. Greubel, Jan 24 2024

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A000700(k)*a(n-k).

From G. C. Greubel, Jan 24 2024: (Start)

G.f.: (1+x)/(2*(1+x) - x*QPochhammer(-1/x; x^2)).

G.f.: 1/( 2 - x^(1/24)*etx(x^2)^2/(eta(x^4)*eta(x)) ), where eta(x) is the Dedekind eta function. (End)