A220418 Express 1 - x - x^2 - x^3 - x^4 - ... as product (1 + g(1)*x) * (1 + g(2)*x^2) *(1 + g(3)*x^3) * ... and use a(n) = - g(n).

1, 1, 2, 3, 6, 8, 18, 27, 54, 84, 186, 296, 630, 1008, 2106, 3711, 7710, 12924, 27594, 48528, 97902, 173352, 364722, 647504, 1340622, 2382660, 4918482, 9052392, 18512790, 33361776, 69273666, 127198287, 258155910, 475568220, 981288906, 1814542704, 3714566310

Offset: 1

Author

Michel Marcus, Dec 14 2012

Keywords

• nonn

Comments

This is the PPE (power product expansion) of A153881 (with offset 0).

When p is prime, a(p) = (2^p-2)/p (A064535).

From Petros Hadjicostas, Oct 04 2019: (Start)

This sequence appears as an example in Gingold and Knopfmacher (1995) starting at p. 1223.

In Section 3 of Gingold and Knopfmacher (1995), it is proved that, if f(z) = Product_{n >= 1} (1 + g(n))*z^n = 1/(Product_{n >= 1} (1 - h(n))*z^n), then g(2*n - 1) = h(2*n - 1) and Sum_{d|n} (1/d)*h(n/d)^d = -Sum_{d|n} (1/d)*(-g(n/d))^d. The same results were proved more than ten years later by Alkauskas (2008, 2009). [If we let a(n) = -g(n), then Alkauskas works with f(z) = Product_{n >= 1} (1 - a(n))*z^n; i.e., a(2*n - 1) = -h(2*n - 1) etc.]

The PPE of 1/(1 - x - x^2 - x^3 - x^4 - ...) is given in A290261, which is also studied in Gingold and Knopfmacher (1995, p. 1234).

(End)

The number of terms in the Zassenhaus formula exponent of order n, as computed by the algorithm by Casas, Murua & Nadinic, is equal to a(n) at least for n = 2..24. - Andrey Zabolotskiy, Apr 09 2023

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..2000
• Giedrius Alkauskas, One curious proof of Fermat's little theorem, arXiv:0801.0805 [math.NT], 2008.
• Giedrius Alkauskas, A curious proof of Fermat's little theorem, Amer. Math. Monthly 116(4) (2009), 362-364.
• Fernando Casas, Ander Murua and Mladen Nadinic, Efficient computation of the Zassenhaus formula, Computer Physics Communications, 183 (2012), 2386-2391; arXiv:1204.0389 [math-ph], 2012.
• H. Gingold, H. W. Gould, and Michael E. Mays, Power Product Expansions, Utilitas Mathematica 34 (1988), 143-161.
• H. Gingold and A. Knopfmacher, Analytic properties of power product expansions, Canad. J. Math. 47 (1995), 1219-1239.
• W. Lang, Recurrences for the general problem.

Crossrefs

Cf. A064535, A147541, A153881, A157162, A170908, A170909, A170910, A170911, A170912, A170913, A170914, A170915, A170916, A170917, A220420, A273866, A290261.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0 or i<1, 1,
      b(n, i-1)+a(i)*b(n-i, min(n-i, i)))
    end:
a:= proc(n) option remember; 2^n-b(n, n-1) end:
seq(a(n), n=1..40);  # Alois P. Heinz, Jun 22 2018

Mathematica

b[n_, i_] := b[n, i] = If[n == 0 || i < 1, 1, b[n, i - 1] + a[i]*b[n - i, Min[n - i, i]]];
a[n_] := a[n] = 2^n - b[n, n - 1] ;
Array[a, 40] (* Jean-François Alcover, Jul 09 2018, after Alois P. Heinz *)

PARI

a(m) = {default(seriesprecision, m+1); gk = vector(m); pol = 1 + sum(n=1, m, -x^n); gk[1] = polcoeff( pol, 1); for (k=2, m, pol = taylor(pol/(1+gk[k-1]*x^(k-1)), x); gk[k] = polcoeff(pol, k, x);); for (k=1, m, print1(-gk[k], ", "););}

Formula

g(1) = -1 and for k > 1, g(k) satisfies Sum_{d|k} (1/d)*(-g(k/d))^d = (2^k - 1)/k, where a(k) = -g(k). - Gevorg Hmayakyan, Jun 05 2016 [Corrected by Petros Hadjicostas, Oct 04 2019. See p. 1224 in Gingold and Knopfmacher (1995).]

From Petros Hadjicostas, Oct 04 2019: (Start)

a(2*n - 1) = A290261(2*n - 1) for n >= 1 because A290261 gives the PPE of 1/(1 - x - x^2 - x^3 - ...) = (1 - x)/(1 - 2*x).

Define (A(m,n): n,m >= 1) by A(m=1,n) = -1 for n >= 1, A(m,n) = 0 for m > n >= 1 (upper triangular), and A(m,n) = A(m-1,n) - A(m-1,m-1) * A(m,n-m+1) for n >= m >= 2. Then a(n) = A(n,n). [Theorem 3 in Gingold et al. (1988).]

(End)

Extensions

Name edited by Petros Hadjicostas, Oct 04 2019