A157162 -id:A157162 - cp's OEIS frontend

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).

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)

A328186 Write 1/(1 + sin x) = Product_{n>=1} (1 + f_n x^n); a(n) = denominator(f_n).

Original entry on oeis.org

1, 1, 6, 6, 120, 360, 5040, 2520, 72576, 1814400, 39916800, 59875200, 1245404160, 21794572800, 1307674368000, 81729648000, 71137485619200, 3201186852864000, 121645100408832000, 12164510040883200, 10218188434341888000, 281000181944401920000, 25852016738884976640000

Offset: 1

Petros Hadjicostas, Oct 06 2019

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/(1 + sin(x)). For that coefficient, we use a modification of a formula by Peter Luschny in the documentation of sequences A099612 and A099617.
Write 1 + sin x = Product_{n>=1} (1 + g_n * x^n). We have A170914(n) = numerator(g_n) and A170915(n) = denominator(g_n).
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.]
Wolfdieter Lang (see the link below) examined inverse power product expansions both for ordinary g.f.'s and for exponential g.f.'s. He connects inverse power product expansions to unital series associated to (infinite dimensional) Witt vectors and to the so-called "Somos transformation".
There are more formulas for f_n and g_n in the references listed below. In all cases, we assume the g.f.'s are unital, i.e., the g.f.'s start with a constant 1.

			f_n = -1, 1, 1/6, 5/6, 19/120, -47/360, 659/5040, 1837/2520, 7675/72576, -154729/1814400, 3578279/39916800, 3984853/59875200, 95259767/1245404160, ...

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 A328191.

Cf. A099612, A099617, A170914, A170915, A279107.

# Calculates the fractions f_n (choose L much larger than M):
PPE := 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/(1 + sin(x));
t0 := series(t1, x, L);
f := []; t2 := t0;
for n to M do
t3 := coeff(t2, x, n);
t2 := series(t2/(1 + t3*x^n), x, L);
f := [op(f), t3];
end do;
end if;
[seq(f[n], n = 1 .. nops(f))];
end proc;
# Calculates the denominators of f_n:
h := map(denom, PPE(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, 2*(-1)^n*I^(n + 2)*PolyLog[-(n + 1), -I]/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 using a program by Jean-François Alcover and a formula from A099612 and A099617 *)

a(2*n + 1) = A170915(2*n + 1) for n >= 0.
Define (A(m,n): n,m >= 1) by A(m=1, n) = 2 * (-1)^n * i^(n + 2) * PolyLog(-(n + 1), -i)/n! for n >= 1 (with i := sqrt(-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 f_n = A(n,n) and thus a(n) = denominator(A(n,n)).
If we write 1 + sin x = Product_{n>=1} (1 + g_n * x^n) and we know (g_n: n >= 1), then f_n = -g_n + Sum_{s|n, s > 1} (1/s) * ((-f_{n/s})^s + (-g_{n/s})^s). This proves of course that f_n = -g_n for n odd.

A328191 Write 1/(1 + sin x) = Product_{n>=1} (1 + f_n x^n); a(n) = numerator(f_n).

Original entry on oeis.org

-1, 1, 1, 5, 19, -47, 659, 1837, 7675, -154729, 3578279, 3984853, 95259767, -1364856587, 85215100151, 46900690817, 4180679675171, -157452879103733, 6399968826052559, 893237028559439, 478887035449041839, -11642446645024341437, 1123931378903214542099, 15392026390952264029

Offset: 1

Petros Hadjicostas, Oct 07 2019

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/(1 + sin(x)). For that coefficient, we use a modification of a formula by Peter Luschny in the documentation of sequences A099612 and A099617.
Write 1 + sin x = Product_{n>=1} (1 + g_n * x^n). We have A170914(n) = numerator(g_n) and A170915(n) = denominator(g_n).
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.]
Wolfdieter Lang (see the link below) examined inverse power product expansions both for ordinary g.f.'s and for exponential g.f.'s. He connects inverse power product expansions to unital series associated to (infinite dimensional) Witt vectors and to the so-called "Somos transformation".
There are more formulas for f_n and g_n in the references listed below. In all cases, we assume the g.f.'s are unital, i.e., the g.f.'s start with a constant 1.

			f_n = -1, 1, 1/6, 5/6, 19/120, -47/360, 659/5040, 1837/2520, 7675/72576, -154729/1814400, 3578279/39916800, 3984853/59875200, 95259767/1245404160, ...

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.

Denominators are in A328186.

Cf. A099612, A099617, A170914, A170915, A279107.

# Calculates the fractions f_n (choose L much larger than M):
PPE := 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/(1 + sin(x));
t0 := series(t1, x, L);
f := []; t2 := t0;
for n to M do
t3 := coeff(t2, x, n);
t2 := series(t2/(1 + t3*x^n), x, L);
f := [op(f), t3];
end do;
end if;
[seq(f[n], n = 1 .. nops(f))];
end proc;
# Calculates the numerators of f_n:
h:=map(numer, PPE(100, 40)); # Petros Hadjicostas, Oct 07 2019 by modifying N. J. A. Sloane's program from A170912 and A170913.

A[m_, n_] :=
  A[m, n] =
   Which[m == 1, 2*(-1)^n*I^(n + 2)*PolyLog[-(n + 1), -I]/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 07 2019 using a program by Jean-François Alcover and a formula from A099612 and A099617 *)

a(2*n + 1) = -A170914(2*n + 1) for n >= 0.
Define (A(m,n): n,m >= 1) by A(m=1, n) = 2 * (-1)^n * i^(n + 2) * PolyLog(-(n + 1), -i)/n! for n >= 1 (with i := sqrt(-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 f_n = A(n,n) and thus a(n) = numerator(A(n,n)).
If we write 1 + sin x = Product_{n>=1} (1 + g_n * x^n) and we know (g_n: n >= 1), then f_n = -g_n + Sum_{s|n, s > 1} (1/s) * ((-f_{n/s})^s + (-g_{n/s})^s). This proves of course that f_n = -g_n for n odd.

A170917 Write sin(x)/x = Product_{n>=1} (1 + g_nx^(2n)); a(n) = denominator(g_n).

Original entry on oeis.org

6, 120, 840, 362880, 14968800, 9340531200, 49037788800, 3201186852864000, 8485288812000, 182467650613248000, 908859963476424960000, 1424498881530396672000000, 10633661572674172032000000, 8289151869130970582384640000000, 1720739115690134518218240000000, 97858575719142221963014963200000000

Offset: 1

N. J. A. Sloane, Jan 30 2010

			-1/6, 1/120, 1/840, 73/362880, 353/14968800, 36499/9340531200, 24257/49037788800, ...

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.
W. Lang, Recurrences for the general problem.

Cf. A170916, A170912-A170915.

t1:=sin(x)/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;
h:=[seq(g[2*n], n=1..nops(g)/2)];
h1:=map(numer, h);
h2:=map(denom, h); # Petros Hadjicostas, Oct 04 2019 by modifying N. J. A. Sloane's program from A170912 and A170913

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

A170916 Write sin(x)/x = Product_{n>=1} (1 + g_nx^(2n)); a(n) = numerator(g_n).

Original entry on oeis.org

-1, 1, 1, 73, 353, 36499, 24257, 302426881, 87721, 348958703, 226786069421, 62199570679633, 62531659610839, 8559230855533306387, 235495453816743509, 2644298730170939345197, 281737789368631676609, 39043444996461526437828311, 6203284926188598376335167

Offset: 1

N. J. A. Sloane, Jan 30 2010

			-1/6, 1/120, 1/840, 73/362880, 353/14968800, 36499/9340531200, 24257/49037788800, ...

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.

Cf. A170917, A170912, A170913, A170914, A170915.

t1:=sin(x)/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;
h:=[seq(g[2*n], n=1..nops(g)/2)];
h1:=map(numer, h);
h2:=map(denom, h); # Petros Hadjicostas, Oct 04 2019 by modifying N. J. A. Sloane's program from A170912 and A170913

A170923 a(n) = denominator of the coefficient c(n) of x^n in sqrt(1+x)/Product_{k=1..n-1} 1 + c(k)*x^k, n = 1, 2, 3, ...

Original entry on oeis.org

2, 8, 8, 128, 32, 512, 128, 32768, 128, 32768, 2048, 2097152, 8192, 2097152, 32768, 2147483648, 131072, 16777216, 524288, 34359738368, 2097152, 8589934592, 8388608, 35184372088832, 524288, 549755813888, 33554432, 562949953421312, 536870912, 35184372088832

Offset: 1

N. J. A. Sloane, Jan 31 2010

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. A170922 (numerators).
Cf. A353583 / A353584 for power product expansion of 1 + tan x.
Cf. A353586 / A353587 for power product expansion of (tan x)/x.

L := 32: g := NULL:
t := series(sqrt(1+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]); # Peter Luschny, May 12 2022

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

A157163 Product_{n>=1} (1 + 2a(n)x^n) = Sum_{k>=0} binomial(2k, k)x^k = 1/sqrt(1 - 4*x), with the central binomial numbers A000984(n).

Original entry on oeis.org

1, 3, 4, 27, 48, 156, 576, 2955, 7168, 27792, 95232, 352188, 1290240, 5105856, 17743872, 77010795, 252641280, 1000224768, 3616800768, 14484040464, 52102692864, 208963943424, 764877471744, 3025006038012, 11258183024640, 44968060784640, 166308918329344

Offset: 1

Wolfdieter Lang, Aug 10 2009

In the original problem 2*a(n) = [2, 6, 8, 54, 96, 312, 1152, 5910, 14336, 55584, 190464, 704376, ...] appears.

			Recurrence I: a(4) = binomial(8, 4)/2 - 2*a(1)*a(3) = 35 - 8 = 27.
Recurrence II: a(4) = (1/2)*(1/2)*(-2*a(2))^2 + (1/2)*(1*cbi(4) - (1/2)*(2*cbi(1)*cbi(3) + 1*cbi(2)^2) + (1/3)*3*cbi(1)^2*cbi(2)) = 27.
Recurrence II (rewritten): a(4)= (1/8)*((-2)^4 + 2*(-2*a(2))^2 + (1/2)*4^4) = 27.

Robert Israel, Table of n, a(n) for n = 1..1665
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.
Art of Problem Solving, Product formula for a generating function
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. A147542 (Fibonacci), A157161 (Catalan).

N:= 100: # for a(1)..a(N)
S:= convert(series(-ln(1-4*x)/2,x,N+1),polynom):
for n from 1 to N do
  a[n]:= coeff(S,x,n)/2;
  S:= S - add((-1)^(k-1)*(2*a[n])^k*x^(k*n)/k, k=1..N/n)
od:
seq(a[n],n=1..N); # Robert Israel, Jan 03 2019

a(n) = if (n==1, 1, (1/(2*n))*((-2*a(1))^n + sumdiv(n, d, if ((d!=1) && (d!=n), d*(-2*a(d))^(n/d), 0)) + 4^n/2)); \\ after 2nd Recurrence II; Michel Marcus, Jul 06 2015

Recurrence I: With FP(n,m) the set of partitions of n with m distinct parts (which could be called fermionic partitions fp):
a(n) = binomial(2*n, n)/2 - Sum_{m=2..maxm(n)} 2^(m-1)*(Sum_{fp from FP(n,m)} (Product_{j=1..m} a(k[j]))), with maxm(n) = A003056(n) and the distinct parts k[j], j = 1..m, of the partition fp of n, n >= 3. Inputs a(1) = 1, a(2) = 3. See the array A008289(n,m) for the cardinality of the set FP(n,m).
Recurrence II: With P(n,m) the set of all partitions of n with m parts, and the multinomial numbers M0 (given for every partition under A048996):
a(n) = (1/2)*Sum_{d|n, 1= 2; a(1) = 1, with cbi(n) = binomial(2*n, n) = A000984(n). The exponents e(j) >= 0 satisfy Sum_{j=1..n} j*e(j) =n and Sum_{j=1..n} e(j) = m. The M0 numbers are m!/(Product_{j=1..n} (e(j))!).
Recurrence II (rewritten, due to email from V. Jovovic, Mar 10 2009):
a(n) = ((-2*a(1))^n + Sum_{d|n, 1

A170922 a(n) = numerator of the coefficient c(n) of x^n in sqrt(1+x)/Product_{k=1..n-1} 1 + c(k)*x^k, n = 1, 2, 3, ...

Original entry on oeis.org

1, -1, 1, -13, 3, -37, 9, -1861, 7, -1491, 93, -81001, 315, -69705, 1083, -63586357, 3855, -438821, 13797, -822684711, 49689, -186369117, 182361, -704368012465, 10485, -10165801275, 619549, -9738266477517, 9256395, -566066862375, 34636833, -140047960975823893

Offset: 1

N. J. A. Sloane, Jan 31 2010

			1/2, -1/8, 1/8, -13/128, 3/32, -37/512, 9/128, -1861/32768, ...

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. A170923 (denominators).
Cf. A353583 / A353584 for power product expansion of 1 + tan x.
Cf. A353586 / A353587 for power product expansion of (tan x)/x.

L := 34: g := NULL:
t := series(sqrt(1+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(numer, [g]); # Peter Luschny, May 12 2022

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

A170924 a(n) = numerator of the coefficient c(n) of x^n in (1/sqrt(1-x))/Product_{k=1..n-1} 1 + c(k)*x^k, n = 1, 2, 3, ...

Original entry on oeis.org

1, 3, 1, 27, 3, 39, 9, 2955, 7, 1737, 93, 88047, 315, 79779, 1083, 77010795, 3855, 488391, 13797, 905252529, 49689, 204066351, 182361, 756251509503, 10485, 10978530465, 619549, 10462007147787, 9256395, 603860858253, 34636833, 150202954242966315

Offset: 1

N. J. A. Sloane, Jan 31 2010

			1/2, 3/8, 1/8, 27/128, 3/32, 39/512, 9/128, 2955/32768, 7/128, ...

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. A170923 (denominators).
Cf. A353583 / A353584 for power product expansion of 1 + tan x.
Cf. A353586 / A353587 for power product expansion of (tan x)/x.

L := 34: g := NULL:
t := series(1/sqrt(1 - 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(numer, [g]); # Peter Luschny, May 12 2022

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

Previous Showing 11-20 of 20 results.

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A328186 Write 1/(1 + sin x) = Product_{n>=1} (1 + f_n x^n); a(n) = denominator(f_n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A328191 Write 1/(1 + sin x) = Product_{n>=1} (1 + f_n x^n); a(n) = numerator(f_n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A170917 Write sin(x)/x = Product_{n>=1} (1 + g_n*x^(2*n)); a(n) = denominator(g_n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

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, ...

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

PARI

Extensions

A170916 Write sin(x)/x = Product_{n>=1} (1 + g_n*x^(2*n)); a(n) = numerator(g_n).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

A170923 a(n) = denominator of the coefficient c(n) of x^n in sqrt(1+x)/Product_{k=1..n-1} 1 + c(k)*x^k, n = 1, 2, 3, ...

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Extensions

A157163 Product_{n>=1} (1 + 2*a(n)*x^n) = Sum_{k>=0} binomial(2*k, k)*x^k = 1/sqrt(1 - 4*x), with the central binomial numbers A000984(n).

Views

Author

Keywords

Comments

Examples

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).

A170917 Write sin(x)/x = Product_{n>=1} (1 + g_nx^(2n)); a(n) = denominator(g_n).

A170916 Write sin(x)/x = Product_{n>=1} (1 + g_nx^(2n)); a(n) = numerator(g_n).

A157163 Product_{n>=1} (1 + 2a(n)x^n) = Sum_{k>=0} binomial(2k, k)x^k = 1/sqrt(1 - 4*x), with the central binomial numbers A000984(n).