A170910 -id:A170910 - 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

A170915 Write 1 + sin x = Product_{n>=1} (1 + g_n * x^n); a(n) = denominator(g_n).

Original entry on oeis.org

1, 1, 6, 6, 120, 120, 5040, 280, 72576, 362880, 39916800, 11975040, 1245404160, 88957440, 1307674368000, 11675664000, 71137485619200, 1067062284288000, 121645100408832000, 101370917007360000, 10218188434341888000, 5109094217170944000, 25852016738884976640000

Offset: 1

N. J. A. Sloane, Jan 30 2010

From Petros Hadjicostas, Oct 06 2019: (Start)
The recurrence about (A(m,n): m,n >= 1) in the Formula section follows from Theorem 3 in Gingold et al. (1988); see also Gingold and Knopfmacher (1995, p. 1222). A(m=1,n) equals the n-th coefficient of the Taylor expansion of 1 + sin(x).
If 1 + sin(x) = 1/Product_{n>=1} (1 + f_n * x^n) (inverse power product expansion), then Gingold and Knopfmacher (1995) and Alkauskas (2008, 2009) proved that f_n = -g_n for n odd, and Sum_{s|n} (-g_{n/s})^s/s = -Sum_{s|n} (-f_{n/s})^s/s. [We caution that different authors may use -g_n for g_n, or -f_n for f_n, or both.] We have A328191(n) = numerator(f_n) and A328186(n) = denominator(f_n).
Wolfdieter Lang (see the link below) examined inverse power product expansions both for ordinary g.f.'s and for exponential g.f.'s.
In all cases, we assume the g.f.'s are unital, i.e., the g.f.'s start with a constant 1.
(End)

			g_n = 1, 0, -1/6, 1/6, -19/120, 19/120, -659/5040, 37/280, -7675/72576, ...

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.
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, 2009.

Numerators are in A170914.

Cf. A170910, A170911, A170912, A170913, A328186, A328191.

# Calculates the fractions g_n (choose L much larger than M):
PPE_sin := proc(L, M)
local t1, t0, g, t2, n, t3;
if L < 2.5*M then print("Choose larger value for L");
else
t1 := 1 + sin(x);
t0 := series(t1, x, L);
g := []; t2 := t0;
for n to M do
t3 := coeff(t2, x, n);
t2 := series(t2/(1 + t3*x^n), x, L);
g := [op(g), t3];
end do;
end if;
[seq(g[n], n = 1 .. nops(g))];
end proc;
# Calculates the denominators of g_n:
h1 := map(denom, PPE_sin(100, 40)); # Petros Hadjicostas, Oct 06 2019 by modifying N. J. A. Sloane's program from A170912 and A170913.

A[m_, n_] :=
  A[m, n] =
   Which[m == 1, (1-(-1)^n)*(-1)^Floor[(n-1)/2]/(2*n!), m > n >= 1, 0, True,
    A[m - 1, n] - A[m - 1, m - 1]*A[m, n - m + 1]];
a[n_] := Denominator[A[n, n]];
a /@ Range[1, 55] (* Petros Hadjicostas, Oct 06 2019, courtesy of Jean-François Alcover *)

From Petros Hadjicostas, Oct 07 2019: (Start)
a(2*n+1) = A328186(2*n+1) for n >= 0.
Define (A(m,n): n,m >= 1) by A(m=1,2*n+1) = (-1)^n/(2*n+1)! for n >= 0, A(m=1,2*n) = 0 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 g_n = A(n,n). (End)

A170914 Write 1 + sin x = Product_{n>=1} (1 + g_n * x^n); a(n) = numerator(g_n).

Original entry on oeis.org

1, 0, -1, 1, -19, 19, -659, 37, -7675, 40043, -3578279, 1123009, -95259767, 7091713, -85215100151, 832857559, -4180679675171, 63804880881241, -6399968826052559, 5697831990097981, -478887035449041839, 252737248941887573, -1123931378903214542099, 35703551772944759

Offset: 1

N. J. A. Sloane, Jan 30 2010

From Petros Hadjicostas, Oct 06 2019: (Start)
The recurrence about (A(m,n): m,n >= 1) in the Formula section follows from Theorem 3 in Gingold et al. (1988); see also Gingold and Knopfmacher (1995, p. 1222). A(m=1,n) equals the n-th coefficient of the Taylor expansion of 1 + sin(x).
If 1 + sin(x) = 1/Product_{n>=1} (1 + f_n * x^n) (inverse power product expansion), then Gingold and Knopfmacher (1995) and Alkauskas (2008, 2009) proved that f_n = -g_n for n odd, and Sum_{s|n} (-g_{n/s})^s/s = -Sum_{s|n} (-f_{n/s})^s/s. [We caution that different authors may use -g_n for g_n, or -f_n for f_n, or both.] We have A328191(n) = numerator(f_n) and A328186(n) = denominator(f_n).
Wolfdieter Lang (see the link below) examined inverse power product expansions both for ordinary g.f.'s and for exponential g.f.'s.
In all cases, we assume the g.f.'s are unital, i.e., the g.f.'s start with a constant 1.
(End)

			g_n = 1, 0, -1/6, 1/6, -19/120, 19/120, -659/5040, 37/280, -7675/72576, ...

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.
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, 2009.

Cf. Denominators are in A170915.

Cf. A170910, A170911, A170912, A170913, A328186, A328191.

# Calculates the fractions g_n (choose L much larger than M):
PPE_sin := proc(L, M)
local t1, t0, g, t2, n, t3;
if L < 2.5*M then print("Choose larger value for L");
else
t1 := 1 + sin(x);
t0 := series(t1, x, L);
g := []; t2 := t0;
for n to M do
t3 := coeff(t2, x, n);
t2 := series(t2/(1 + t3*x^n), x, L);
g := [op(g), t3];
end do;
end if;
[seq(g[n], n = 1 .. nops(g))];
end proc;
# Calculates the numerators of g_n:
h1 := map(numer, PPE_sin(100, 40)); # Petros Hadjicostas, Oct 06 2019 by modifying N. J. A. Sloane's program from A170912 and A170913.

A[m_, n_] :=
  A[m, n] =
   Which[m == 1, (1-(-1)^n)*(-1)^Floor[(n-1)/2]/(2*n!), m > n >= 1, 0, True,
    A[m - 1, n] - A[m - 1, m - 1]*A[m, n - m + 1]];
a[n_] := Numerator[A[n, n]];
a /@ Range[1, 55] (* Petros Hadjicostas, Oct 06 2019, courtesy of Jean-François Alcover *)

From Petros Hadjicostas, Oct 07 2019: (Start)
a(2*n+1) = -A328191(2*n+1) for n >= 0.
Define (A(m,n): n,m >= 1) by A(m=1,2*n+1) = (-1)^n/(2*n+1)! for n >= 0, A(m=1,2*n) = 0 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 g_n = A(n,n). (End)

A170911 Write exp(-x) = Product_{n>=1} (1 + g_n x^n); a(n) = denominator(g_n).

Original entry on oeis.org

1, 2, 3, 8, 5, 72, 7, 128, 81, 800, 11, 13824, 13, 6272, 30375, 32768, 17, 419904, 19, 20480000, 750141, 247808, 23, 1528823808, 15625, 1384448, 1594323, 5035261952, 29, 30233088000000, 31, 2147483648, 235782657, 37879808, 1313046875, 240734712102912, 37, 189267968

Offset: 1

N. J. A. Sloane, Jan 30 2010

			-1, 1/2, 1/3, 3/8, 1/5, 13/72, 1/7, 27/128, 8/81, 91/800, 1/11, ...

H. Gingold, H. W. Gould, and Michael E. Mays, Power Product Expansions, Utilitas Mathematica 34 (1988), 143-161.

Cf. A170910 (numerators).

L:=100; t1:=exp(-x); t0:=series(t1,x,L): g:=[]; M:=40; t2:=t0:
for n from 1 to M do t3:=coeff(t2,x,n); t2:=series(t2/(1+t3*x^n),x,L); g:=[op(g),t3]; od: g;

A170919 a(n) = denominator of the coefficient c(n) of x^n in (tan x)/Product_{k=1..n-1} 1 + c(k)*x^k, n = 1, 2, 3, ...

Original entry on oeis.org

1, 1, 3, 3, 5, 45, 105, 315, 2835, 14175, 5775, 467775, 6081075, 2837835, 212837625, 70945875, 3618239625, 97692469875, 206239658625, 9280784638125, 1031198293125, 142924083427125, 322279795963125, 101111706320625, 136968913284328125, 161872352063296875

Offset: 1

N. J. A. Sloane, Jan 30 2010

			1, -1, 7/3, -14/3, 54/5, -1112/45, 6574/105, -48488/315, 1143731/2835, ...

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, Canad. J. Math. 47 (1995), 1219-1239.
Wolfdieter Lang, Recurrences for the general problem.

Cf. A170918 (numerators), A170910-A170917.
Cf. A353583 / A353584 for power product expansion of 1 + tan x.
Cf. A353586 / A353587 for power product expansion of (tan x)/x.

L := 28: g := NULL:
t := series(tan(x), x, L):
for n from 1 to L-2 do
   c := coeff(t, x, n);
   t := series(t/(1 + c*x^n), x, L);
   g := g, c;
od: map(denom, [g]); # Based on Maple in A170918. - Peter Luschny, Oct 05 2019

Following a suggestion from Ilya Gutkovskiy, name corrected so that it matches the data by Peter Luschny, May 12 2022

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)

A170918 a(n) = numerator of the coefficient c(n) of x^n in (tan x)/Product_{0 < k < n} 1 + c(k)*x^k, n = 1, 2, 3, ...

Original entry on oeis.org

1, -1, 7, -14, 54, -1112, 6574, -48488, 1143731, -14813072, 16252211, -3500388967, 125127865048, -158589803803, 33133618166566, -30512906279732, 4378989933312913, -330336346477870319, 1981395373839282068, -251479418962683770473, 79893293800974935213, -31493610597939643431532

Offset: 1

N. J. A. Sloane, Jan 30 2010

			1, -1, 7/3, -14/3, 54/5, -1112/45, 6574/105, -48488/315, 1143731/2835, ...

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, Canad. J. Math. 47 (1995), 1219-1239.
Wolfdieter Lang, Recurrences for the general problem.

Cf. A170919 (denominators), A170910-A170917.
Cf. A353583 / A353584 for power product expansion of 1 + tan x.
Cf. A353586 / A353587 for power product expansion of (tan x)/x.

t1:=tan(x);
L:=100;
t0:=series(t1,x,L):
g:=[]; M:=40; t2:=t0:
for n from 1 to M do
t3:=coeff(t2,x,n); t2:=series(t2/(1+t3*x^n),x,L); g:=[op(g),t3];
od:
g;
g1:=map(numer,g);
g2:=map(denom,g);

t=tan(x+O(x)^25); vector(#t,n,c=polcoef(t,n);t/=1+c*x^n;numerator(c)) \\ M. F. Hasler, May 07 2022

Following a suggestion from Ilya Gutkovskiy, name corrected so that it matches the data by M. F. Hasler, May 07 2022

A354016 Product_{n>=1} (1 + x^n/n!)^a(n) = exp(x).

Original entry on oeis.org

1, 1, -2, 9, -24, 70, -720, 5985, -39200, 337176, -3628800, 40907790, -479001600, 6128488080, -87104969952, 1318070979225, -20922789888000, 354250929192160, -6402373705728000, 121882099274319384, -2432849766865689600, 51041047393559059200

Offset: 1

Ilya Gutkovskiy, May 14 2022

sign

Cf. A006973, A137852, A170910, A170911, A353822.

nn = 22; f[x_] := Product[(1 + x^n/n!)^a[n], {n, 1, nn}]; sol = SolveAlways[0 == Series[f[x] - Exp[x], {x, 0, nn}], x]; Table[a[n], {n, 1, nn}] /. sol // Flatten
a[1] = 1; a[n_] := a[n] = (n - 1)! ((-1)^n + Sum[d (-d!)^(-n/d) a[d], {d, Divisors[n] ~ Complement ~ {1, n}}]); Table[a[n], {n, 1, 22}]

a(1) = 1; a(n) = (n-1)! * ((-1)^n + Sum_{d|n, 1 < d < n} d * (-d!)^(-n/d) * a(d)).

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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A170915 Write 1 + sin x = Product_{n>=1} (1 + g_n * x^n); a(n) = denominator(g_n).

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A170914 Write 1 + sin x = Product_{n>=1} (1 + g_n * x^n); a(n) = numerator(g_n).

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Maple

Mathematica

Formula

A170911 Write exp(-x) = Product_{n>=1} (1 + g_n x^n); a(n) = denominator(g_n).

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Maple

A170919 a(n) = denominator of the coefficient c(n) of x^n in (tan x)/Product_{k=1..n-1} 1 + c(k)*x^k, n = 1, 2, 3, ...

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Maple

Extensions

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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Mathematica

PARI

Formula

A170918 a(n) = numerator of the coefficient c(n) of x^n in (tan x)/Product_{0 < k < n} 1 + c(k)*x^k, n = 1, 2, 3, ...

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Extensions

A354016 Product_{n>=1} (1 + x^n/n!)^a(n) = exp(x).

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).