A170908 -id:A170908 - cp's OEIS frontend

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

Michel Marcus, Dec 14 2012

nonn

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

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.

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

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

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 *)

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], ", "););}

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)

Name edited by Petros Hadjicostas, Oct 04 2019

A220420 Express the Sum_{n>=0} p(n)x^n, where p(n) is the partition function, as a product Product_{k>=1} (1 + a(k)x^k).

Original entry on oeis.org

1, 2, 1, 4, 1, 0, 1, 14, 1, -4, 1, -8, 1, -16, 1, 196, 1, -54, 1, -92, 1, -184, 1, 144, 1, -628, 1, -1040, 1, -2160, 1, 41102, 1, -7708, 1, -12932, 1, -27592, 1, 54020, 1, -98496, 1, -173720, 1, -364720, 1, 853624, 1, -1341970, 1, -2383916, 1, -4918536, 1

Offset: 1

Michel Marcus, Dec 14 2012

sign

This is the PPE (power product expansion) of A000041.
When n is odd, a(n) = 1.
When n is even, a(n) = 2, 4, 0, 14, -4, -8, -16, 196, -54, -92, -184, 144, -628, -1040, -2160, 41102, ...
Alkauskas (2016, Problem 3, p. 3) conjectured that a(8*k+2), a(8*k+4), and a(8*k+6) are all negative, and a(8*k) is positive for k >= 1. [This statement is not wholly true for k = 0.] - Petros Hadjicostas, Oct 07 2019

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.
Giedrius Alkauskas, Algebraic functions with Fermat property, eigenvalues of transfer operator and Riemann zeros, and other open problems, arXiv:1609.09842 [math.NT], 2016.
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, Canadian Journal of Mathematics 47(6) (1995), 1219-1239.
W. Lang, Recurrences for the general problem.

Cf. A000041, A147541, A170908, A170909, A170910, A170911, A170912, A170913, A170914, A170915, A170916, A170917, A220418, A290261.

terms = 55; sol[0] = {};
sol[m_] := sol[m] = Join[sol[m - 1], If[OddQ[m], {a[m] -> 1}, First @ Solve[Thread[Table[PartitionsP[n], {n, 0, m}] == CoefficientList[ (Product[1 + a[n]*x^n, {n, 1, m}] /. sol[m - 1]) + O[x]^(m + 1), x]]]]];
Array[a, terms] /. sol[terms] (* Jean-François Alcover, Dec 06 2018, corrected Oct 03 2019 *)
(* Second program: *)
A[m_, n_] := A[m, n] = Which[m == 1, PartitionsP[n], m > n >= 1, 0, True, A[m - 1, n] - A[m - 1, m - 1]*A[m, n - m + 1] ];
a[n_] := A[n, n];
a /@ Range[1, 55] (* Jean-François Alcover, Oct 03 2019, using the formula given by Petros Hadjicostas *)

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

From Petros Hadjicostas, Oct 04 2019: (Start)
Define (A(m,n): n,m >= 1) by A(m=1,n) = p(n) = A000041(n) 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).]
a(n) = Sum_{s|n} s/n + Sum_{s|n, s > 1} (-a(n/s))^s/s. [Eq. (1) in Alkauskas (2008, 2009).]
(End)

A170909 Denominators in Taylor series expansion of Product_{n >= 1} (1+x^n/n!).

Original entry on oeis.org

1, 1, 2, 3, 24, 15, 360, 5040, 40320, 11340, 1814400, 39916800, 47900160, 6227020800, 7925299200, 1307674368000, 20922789888000, 11115232128000, 3201186852864000, 24329020081766400, 48658040163532800, 51090942171709440000, 1124000727777607680000, 5170403347776995328000

Offset: 0

N. J. A. Sloane, Jan 30 2010

			1 + x + (1/2)*x^2 + (2/3)*x^3 + (5/24)*x^4 + (2/15)*x^5 + (41/360)*x^6 + (169/5040)*x^7 + ...

H. Gingold, H. W. Gould, and M. E. Mays, Power product expansions, Util. Math., 34 (1988), 143-161.

Cf. A170908.

nmax=23; Denominator[CoefficientList[Series[Product[ (1+x^n/n!),{n,nmax}],{x,0,nmax}],x]] (* Stefano Spezia, Jun 24 2024 *)

a(23) from Stefano Spezia, Jun 24 2024

A170920 Write xcot(x) = Product_{n>=1} (1 + g_nx^(2*n)); a(n) = numerator(g_n).

Original entry on oeis.org

-1, -1, -1, -16, -91, -58844, -73267, -1196588, -49830764, -1715330699, -35249288479, -374085503198546, -732125336837021, -779432268293710651, -30015706187367326893, -183998031852529374082, -46789354983174555461, -115977125342651266593554, -248130101882943187003954597, -13171311382437535379302071714878

Offset: 1

N. J. A. Sloane, Jan 31 2010

			-1/3, -1/45, -1/105, -16/4725, -91/66825, -58844/127702575, -73267/383107725, ...

Cf. A170921, A170908-A170919.

t1:=x*cot(x);
L:=100;
t0:=series(t1, x, L);
g:=[];
M:=20; # number of terms to get
t2:=t0:
for n from 1 to M do
t3:=coeff(t2, x, 2*n); t2:=series(t2/(1+t3*x^(2*n)), x, L); g:=[op(g), t3];
od:
g;
g1:=map(numer, g);
g2:=map(denom, g);

Corrected definition and terms - N. J. A. Sloane, Oct 04 2019 (thanks to Petros Hadjicostas for pointing out that something was wrong).

cp's OEIS Frontend

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).

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A220420 Express the Sum_{n>=0} p(n)*x^n, where p(n) is the partition function, as a product Product_{k>=1} (1 + a(k)*x^k).

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A170909 Denominators in Taylor series expansion of Product_{n >= 1} (1+x^n/n!).

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A170920 Write x*cot(x) = Product_{n>=1} (1 + g_n*x^(2*n)); a(n) = numerator(g_n).

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Programs

Maple

Extensions

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).

A220420 Express the Sum_{n>=0} p(n)x^n, where p(n) is the partition function, as a product Product_{k>=1} (1 + a(k)x^k).

A170920 Write xcot(x) = Product_{n>=1} (1 + g_nx^(2*n)); a(n) = numerator(g_n).