A317536 Expansion of 1/(1 + 1/(1 - x) - Product_{k>=1} (1 + x^k)).

1, 0, 0, 1, 1, 2, 4, 6, 10, 18, 30, 50, 86, 145, 245, 417, 705, 1193, 2024, 3427, 5804, 9836, 16660, 28220, 47811, 80991, 137197, 232423, 393729, 666982, 1129898, 1914078, 3242495, 5492898, 9305130, 15763154, 26703273, 45236138, 76631348, 129815818, 219911870, 372537244, 631089250

Offset: 0

Author

Ilya Gutkovskiy, Jul 30 2018

Keywords

• nonn

Comments

Invert transform of A111133.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..4371
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000009, A050342, A089259, A111133, A270995, A279785, A299106, A304969, A305651, A317535.

Programs

Maple

seq(coeff(series(1/(1+1/(1-x)-mul(1+x^k,k=1..n)), x,n+1),x,n),n=0..50); # Muniru A Asiru, Jul 30 2018
# second Maple program:
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
     `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
    end:
a:= proc(n) option remember; `if`(n<1, 1,
      add(a(n-i)*(b(i)-1), i=1..n))
    end:
seq(a(n), n=0..42);  # Alois P. Heinz, Dec 16 2022

Mathematica

nmax = 42; CoefficientList[Series[1/(1 + 1/(1 - x) - Product[(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 42; CoefficientList[Series[1/(1 - Sum[(PartitionsQ[k] - 1) x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[(PartitionsQ[k] - 1) a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 42}]

Formula

G.f.: 1/(1 - Sum_{k>=1} A111133(k)*x^k).