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

1,3

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.

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)

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 *)

Prog

(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], ", "); ); }

Crossrefs

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

Sequence in context: A110448 A005508 A022542 * A064450 A217137 A248824

Adjacent sequences: A220415 A220416 A220417 * A220419 A220420 A220421

Keyword

nonn

Author

Michel Marcus, Dec 14 2012

Extensions

Name edited by Petros Hadjicostas, Oct 04 2019

Status

approved