A258406 -id:A258406 - cp's OEIS frontend

A002107 Expansion of Product_{k>=1} (1 - x^k)^2.

1, -2, -1, 2, 1, 2, -2, 0, -2, -2, 1, 0, 0, 2, 3, -2, 2, 0, 0, -2, -2, 0, 0, -2, -1, 0, 2, 2, -2, 2, 1, 2, 0, 2, -2, -2, 2, 0, -2, 0, -4, 0, 0, 0, 1, -2, 0, 0, 2, 0, 2, 2, 1, -2, 0, 2, 2, 0, 0, -2, 0, -2, 0, -2, 2, 0, -4, 0, 0, -2, -1, 2, 0, 2, 0, 0, 0, -2

Offset: 0

N. J. A. Sloane

Number of partitions of n into an even number of distinct parts minus number of partitions of n into an odd number of distinct parts, with 2 types of each part. E.g., for n=4, we consider k and k* to be different versions of k and so we have 4, 4*, 31, 31*, 3*1, 3*1*, 22*, 211*, 2*11*. The even partitions number 5 and the odd partitions number 4, so a(4)=5-4=1. - Jon Perry, Apr 04 2004
Also, number of partitions of n into parts of -2 different kinds (based upon formal analogy). - Michele Dondi (blazar(AT)lcm.mi.infn.it), Jun 29 2004
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 68 of the 74 eta-quotients listed in Table I of Martin (1996).

			G.f. = 1 - 2*x - x^2 + 2*x^3 + x^4 + 2*x^5 - 2*x^6 - 2*x^8 - 2*x^9 + x^10 + ...
G.f. = q - 2*q^13 - q^25 + 2*q^37 + q^49 + 2*q^61 - 2*q^73 - 2*q^97 - 2*q^109 + ...

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
G. E. Andrews, Advanced problems 6562, Amer. Math. Monthly 94, 1987.
M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
S. Cooper, M. D. Hirschhorn and R. Lewis, Powers of Euler's Product and Related Identities, The Ramanujan Journal, Vol. 4 (2), 137-155 (2000).
S. R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
J. W. L. Glaisher, On the square of Euler's series, Proc. London Math. Soc., 21 (1889), 182-194.
J. T. Joichi, Hecke-Rogers, Andrews identities; combinatorial proofs, Discrete Mathematics, Vol. 84, Issue 3, 1990, pp. 255-259.
Victor G. Kac and Dale H. Peterson, Infinite-Dimensional Lie Algebras, Theta Functions and Modular Forms, Advances in Mathematics (1984), 53. 125-264, see page 261, (5.19).
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
V. Kotesovec, The integration of q-series
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Tim Silverman, Counting Cliques in Finite Distant Graphs, arXiv preprint arXiv:1612.08085 [math.CO], 2016.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Cf. A000712 (reciprocal of g.f.), A010815, A010816, A258406.

Powers of Euler's product: A000594, A000727 - A000731, A000735, A000739, A010815 - A010840.

# DedekindEta is defined in A000594.
A002107List(len) = DedekindEta(len, 2)
A002107List(78) |> println # Peter Luschny, Mar 09 2018

Basis( CuspForms( Gamma1(144), 1), 926) [1]; /* Michael Somos, May 17 2015 */

A010816 := proc (n); if frac(sqrt(8*n+1)) = 0 then (-1)^((1/2)*isqrt(8*n+1)-1/2)*isqrt(8*n+1) else 0 end if; end proc:
N := 10:
a := proc (n) option remember; if n < 0 then 0 else A010816(n) + add( (-1)^(k+1)*a(n - (1/2)*k*(3*k-1) ), k = -N..-1) + add( (-1)^(k+1)*a(n - (1/2)*k*(3*k-1) ), k = 1..N) end if; end proc:
seq(a(n), n = 0..100); # Peter Bala, Apr 06 2022

terms = 78; Clear[s]; s[n_] := s[n] = Product[(1 - x^k)^2, {k, 1, n}] // Expand // CoefficientList[#, x]& // Take[#, terms]&; s[n = 10]; s[n = 2*n]; While[s[n] != s[n - 1], n = 2*n]; A002107 = s[n] (* Jean-François Alcover, Jan 17 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2, {x, 0, n}]; (* Michael Somos, Jan 31 2015 *)
a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, n}]^2, {x, 0, n}]; (* Michael Somos, Jan 31 2015 *)

{a(n) = my(A, p, e, x); if( n<0, 0, n = 12*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p<5, 0, p%12>1, if( e%2, 0, (-1)^((p%12==5) * e/2)), for( i=1, sqrtint(p\9), if( issquare(p - 9*i^2), x=i; break)); (e + 1) * (-1)^(e*x))))}; /* Michael Somos, Aug 30 2006 */

{a(n) = if( n<0, 0, polcoeff( eta(x + x * O(x^n))^2, n))}; /* Michael Somos, Aug 30 2006 */

Vec(eta(x)^2) \\ Charles R Greathouse IV, Apr 22 2016

Expansion of q^(-1/12) * eta(q)^2 in powers of q. - Michael Somos, Mar 06 2012
Euler transform of period 1 sequence [ -2, ...]. - Michael Somos, Mar 06 2012
a(n) = b(12*n + 1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 7, 11 (mod 12), b(p^e) = (-1)^(e/2) * (1 + (-1)^e) / 2 if p == 5 (mod 12), b(p^e) = (e + 1) * (-1)^(e*x) if p == 1 (mod 12) where p = x^2 + 9*y^2. - Michael Somos, Sep 16 2006
Convolution inverse of A000712.
a(0) = 1, a(n) = -(2/n)*Sum{k = 0..n-1} a(k)*sigma_1(n-k). - Joerg Arndt, Feb 05 2011
Expansion of f(-x)^2 in powers of x where f() is a Ramanujan theta function. - Michael Somos, May 17 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 12 (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, May 17 2015
a(n) = Sum_{k=0..n} A010815(k)*A010815(n-k); self convolution of A010815. - Gevorg Hmayakyan, Sep 18 2016
G.f.: Sum_{m, n in Z, n >= 2*|m|} (-1)^n * x^((3*(2*n + 1)^2 - (6*m + 1)^2)/24). - Seiichi Manyama, Oct 01 2016
G.f.: exp(-2*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018
From Peter Bala, Jan 02 2021: (Start)
For prime p congruent to 5, 7 or 11 (mod 12), a(n*p^2 + (p^2 - 1)/12) = e*a(n), where e = 1 if p == 7 or 11 (mod 12) and e = -1 if p == 5 (mod 12).
If n and p are coprime then a(n*p + (p^2 - 1)/12) = 0. See Cooper et al., Theorem 1. (End)
With the convention that a(n) = 0 for n < 0 we have the recurrence a(n) = A010816(n) + Sum_{k a nonzero integer} (-1)^(k+1)*a(n - k*(3*k-1)/2), where A010816(n) = (-1)^m*(2*m+1) if n = m*(m + 1)/2, with m positive, is a triangular number else equals 0. For example, n = 10 = (4*5)/2 is a triangular number, A010816(10) = 9, and so a(10) = 9 + a(9) + a(8) - a(5) - a(3) = 9 - 2 - 2 - 2 - 2 = 1. - Peter Bala, Apr 06 2022

A258232 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k) dx.

Original entry on oeis.org

3, 6, 8, 4, 1, 2, 5, 3, 5, 9, 3, 1, 4, 3, 3, 6, 5, 2, 3, 2, 1, 3, 1, 6, 5, 9, 7, 3, 2, 7, 8, 5, 1, 0, 1, 5, 0, 1, 4, 2, 4, 1, 3, 0, 3, 9, 2, 8, 8, 1, 9, 9, 6, 8, 3, 0, 3, 6, 1, 5, 8, 0, 6, 6, 8, 2, 8, 1, 4, 7, 3, 0, 0, 8, 8, 9, 0, 3, 4, 3, 9, 2, 9, 8, 9, 0, 6, 3, 4, 4, 2, 4, 2, 4, 1, 4, 9, 9, 2, 1, 7, 6, 7, 1, 2, 8

Offset: 0

Vaclav Kotesovec, May 24 2015

			0.3684125359314336523213165973278510150142413039288199683036158...

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, p. 424.

Jonathan M. Borwein and Peter B. Borwein, Strange series and high precision fraud, The American Mathematical Monthly, Vol. 99, No. 7 (1992), pp. 622-640; alternative link.
Martin Klazar, What is an answer? - remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Vaclav Kotesovec, The integration of q-series.

Cf. A000120, A010815, A029931, A242168, A258229, A258230, A258408.

Cf. A258406 (m=2), A258407 (m=3), A258404 (m=4), A258405 (m=5).

evalf(8*sqrt(3/23)*Pi*sinh(sqrt(23)*Pi/6)/(2*cosh(sqrt(23)*Pi/3)-1), 123);
evalf(Sum((-1)^n/((3*n-1)*n/2 + 1), n=-infinity..infinity), 123);

RealDigits[N[8*Sqrt[3/23]*Pi*Sinh[Sqrt[23]*Pi/6] / (2*Cosh[Sqrt[23]*Pi/3]-1),120]][[1]]

8*Pi*sqrt(3/23) * sinh(sqrt(23)*Pi/6) / (2*cosh(sqrt(23)*Pi/3) - 1) \\ Michel Marcus, Nov 28 2018

Equals 8*Pi*sqrt(3/23) * sinh(sqrt(23)*Pi/6) / (2*cosh(sqrt(23)*Pi/3) - 1).
From Amiram Eldar, Feb 04 2024: (Start)
Equals 2 * Sum_{k=-oo..oo} (-1)^k/(3*k^2 + k + 2).
Equals Sum_{k>=0} (-1)^A000120(k)/(A029931(k)+1) (Borwein and Borwein, 1992). (End)

A258414 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^(24*k)) dx.

Original entry on oeis.org

9, 4, 9, 7, 0, 3, 1, 2, 6, 2, 9, 4, 0, 0, 9, 3, 9, 5, 2, 6, 3, 4, 9, 8, 4, 9, 1, 7, 4, 5, 7, 4, 1, 5, 1, 5, 8, 7, 3, 6, 5, 1, 9, 5, 0, 9, 0, 9, 6, 9, 2, 9, 4, 4, 8, 8, 0, 9, 1, 7, 6, 5, 4, 3, 6, 8, 3, 0, 5, 1, 9, 5, 5, 6, 8, 7, 9, 2, 8, 8, 1, 7, 2, 6, 0, 0, 6, 8, 0, 3, 2, 8, 4, 8, 3, 5, 3, 5, 0, 1, 6, 8, 7, 2, 9, 0

Offset: 0

Vaclav Kotesovec, May 29 2015

Integral_{x=-1..1} Product_{k>=1} (1-x^(24*k)) dx = Pi^2/(3*sqrt(3)) = 1.89940625258801878... . - Vaclav Kotesovec, Jun 02 2015

Equals the value of the Dirichlet L-series of a non-principal character modulo 12 (A110161) at s=2. - Jianing Song, Nov 16 2019

			0.9497031262940093952634984917457415158736519509096929448809176543683...

M. W. Coffey, Summatory relations and prime products for the Stieltjes constants and other related results, arXiv:1701.07064 [math.NT], 2017, eq. (2.1).
Vaclav Kotesovec, The integration of q-series.

Cf. A258232, A258406, A258408.
Cf. A196530, A110161.
Cf. A346451, A346450, A346449.

Maple
```
evalf(Pi^2/(6*sqrt(3)), 120);
```

RealDigits[Pi^2/(6*Sqrt[3]),10,120][[1]]
N[Sum[(-1)^n/(12*n*(3n-1)+1),{n,-Infinity,Infinity}],105]

Equals Pi^2/(6*sqrt(3)).
Equals Sum_{k>=1} A110161(n)/k^2 = Sum_{k>=1} Kronecker(12,k)/k^2. - Jianing Song, Nov 16 2019
Equals -Integral_{x=0..oo} log(x)/(x^6 + 1) dx. - Amiram Eldar, Aug 12 2020
Equals 1 + Sum_{k>=1} ( (-1)^k/(6*k-1)^2 + (-1)^k/(6*k+1)^2 ). - Sean A. Irvine, Jul 18 2021
Equals 1/(Product_{p prime == 1 or 11 (mod 12)} (1 - 1/p^2) * Product_{p prime == 5 or 7 (mod 12)} (1 + 1/p^2)). - Amiram Eldar, Dec 17 2023

A258404 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^4 dx.

Original entry on oeis.org

1, 6, 1, 8, 2, 0, 2, 4, 2, 2, 9, 4, 8, 5, 6, 5, 6, 1, 8, 0, 2, 6, 1, 3, 3, 4, 9, 8, 5, 7, 8, 6, 5, 3, 4, 3, 1, 3, 0, 6, 8, 5, 7, 8, 2, 8, 8, 0, 1, 8, 9, 9, 0, 3, 9, 8, 0, 4, 2, 9, 4, 5, 3, 5, 7, 9, 5, 3, 4, 1, 5, 3, 8, 0, 4, 3, 7, 1, 4, 8, 9, 6, 8, 8, 5, 3, 3, 7, 1, 2, 9, 9, 2, 1, 5, 8, 5, 4, 4, 8, 5, 2, 1, 8, 9, 9

Offset: 0

Vaclav Kotesovec, May 29 2015

			0.16182024229485656180261334985786534313068578288018990398...

Vaclav Kotesovec, The integration of q-series

Cf. A000727, A258232, A258406, A258407, A258404, A258405.

evalf(Sum((2*Pi*(-1)^m / cosh(sqrt(7 - 4*m + 12*m^2)*Pi/2)), m=-infinity..infinity), 120); # Vaclav Kotesovec, Dec 04 2015

nmax=200; p=1; q4=Table[PrintTemporary[n]; p=Expand[p*(1-x^n)^4]; Total[CoefficientList[p,x]/Range[1,Exponent[p,x]+1]],{n,1,nmax}]; q4n=N[q4,1000]; Table[SequenceLimit[Take[q4n,j]],{j,Length[q4n]-100,Length[q4n],10}]
NSum[2*(-1)^m*Pi/Cosh[Sqrt[7 - 4*m + 12*m^2]*Pi/2], {m, -Infinity, Infinity}, WorkingPrecision -> 120, NSumTerms -> 100] (* Vaclav Kotesovec, Dec 04 2015 *)
RealDigits[NIntegrate[QPochhammer[x]^4, {x, 0, 1}, WorkingPrecision -> 120], 10, 106][[1]] (* Vaclav Kotesovec, Oct 10 2023 *)

default(realprecision, 93);
b(n) = cosh(sqrt(7 - 4*n + 12*n^2)*Pi/2);
2*Pi*(1/b(0) + sumalt(n=1, (-1)^n*(1/b(n) + 1/b(-n)))) \\ Gheorghe Coserea, Sep 26 2018

Sum_{m = -infinity..infinity} (2*Pi*(-1)^m / cosh(sqrt(7 - 4*m + 12*m^2)*Pi/2)). - Vaclav Kotesovec, Dec 04 2015

More digits from Vaclav Kotesovec, Oct 10 2023

A258405 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^5 dx.

Original entry on oeis.org

1, 3, 7, 8, 0, 1, 0, 7, 0, 8, 4, 6, 5, 5, 4, 6, 4, 2, 8, 4, 5, 3, 8, 6, 1, 3, 1, 4, 0, 2, 1, 9, 3, 8, 4, 3, 0, 8, 4, 5, 2, 2, 5, 4, 1, 2, 3, 2, 6, 2, 5, 9, 8, 2, 6, 8, 3, 9, 3, 7, 0, 0, 3, 7, 0, 9, 2, 4, 8, 6, 3, 1, 0, 7, 7, 3, 1, 8, 1, 7, 0, 4, 8, 9, 3, 6, 2, 9, 1, 7, 6, 9, 8, 5, 9, 2, 4, 3, 4, 4, 1, 4

Offset: 0

Vaclav Kotesovec, May 29 2015

			0.137801070846554642845386131402193843084522541232625982683937003709248631...

Vaclav Kotesovec, The integration of q-series

Cf. A000728, A258232, A258406, A258407, A258404, A258405.

evalf(Sum(Sum(2*Pi*(-1)^(h+m) / cosh(sqrt(7 - 4*h + 12*h^2 - 4*m + 12*m^2)*Pi/2), m=-infinity..infinity), h=-infinity..infinity), 120); # Vaclav Kotesovec, Dec 04 2015

nmax=200; p=1; q5=Table[PrintTemporary[n]; p=Expand[p*(1-x^n)^5]; Total[CoefficientList[p,x]/Range[1,Exponent[p,x]+1]],{n,1,nmax}]; q5n=N[q5,1000]; Table[SequenceLimit[Take[q5n,j]],{j,Length[q5n]-100,Length[q5n],10}]
nterms = 40; N[Sum[Sum[2*Pi*(-1)^(h+m) / Cosh[Sqrt[7 - 4*h + 12*h^2 - 4*m + 12*m^2]*Pi/2], {m, -nterms, nterms}], {h, -nterms, nterms}], 100] (* Vaclav Kotesovec, Dec 04 2015 *)
RealDigits[NIntegrate[QPochhammer[x]^5, {x, 0, 1}, WorkingPrecision -> 120], 10, 106][[1]] (* Vaclav Kotesovec, Oct 10 2023 *)

Sum_{m = -infinity..infinity} (Sum_{h = -infinity..infinity} (2*Pi*(-1)^(h+m) / cosh(sqrt(7 - 4*h + 12*h^2 - 4*m + 12*m^2)*Pi/2))). - Vaclav Kotesovec, Dec 04 2015

A258407 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^3 dx.

Original entry on oeis.org

1, 9, 6, 8, 8, 0, 6, 1, 5, 3, 1, 4, 5, 8, 8, 9, 7, 5, 3, 5, 3, 3, 5, 1, 3, 5, 8, 4, 7, 6, 9, 6, 6, 6, 8, 2, 9, 6, 6, 7, 3, 4, 3, 1, 7, 8, 3, 9, 1, 7, 5, 7, 5, 8, 6, 0, 9, 3, 3, 5, 7, 0, 6, 2, 6, 8, 9, 9, 0, 1, 5, 1, 1, 1, 1, 0, 5, 6, 2, 0, 9, 2, 2, 2, 9, 0, 5, 1, 0, 6, 0, 2, 7, 8, 3, 7, 4, 5, 6, 7, 3, 5, 4, 1, 8, 3

Offset: 0

Vaclav Kotesovec, May 29 2015

In general, Integral_{x=0..1} Product_{k>=1} (1-x^(m*k))^3 dx = Sum_{n>=0} (-1)^n * (2*n+1) / (m*n*(n+1)/2 + 1) is equal to

if 0

if m = 8: Pi/4

if m > 8: 2*Pi / (m * cos((Pi/2)*sqrt(1-8/m)))

Special values: m=4: Pi/(2*cosh(Pi/2)), m=9: 4*Pi/(9*sqrt(3)).

---

Integral_{x=-1..1} Product_{k>=1} (1-x^k)^3 dx = 2*Pi*(1 + sqrt(2) * cosh(sqrt(7)*Pi/4)) / cosh(sqrt(7)*Pi/2) = 1.32639350417409769439126... . - Vaclav Kotesovec, Jun 02 2015

			0.1968806153145889753533513584769666829667343178391757586093357...

Vaclav Kotesovec, The integration of q-series

Cf. A010816, A258232, A258406, A258404, A258405.

evalf(2*Pi/cosh(sqrt(7)*Pi/2), 120);
evalf(Sum((-1)^n * (2*n+1) / (n*(n+1)/2 + 1), n=0..infinity), 120);

RealDigits[2*Pi*Sech[(Sqrt[7]*Pi)/2],10,105][[1]]

2*Pi/cosh((sqrt(7)*Pi)/2) \\ Stefano Spezia, Aug 23 2025

Equals 2*Pi/cosh(sqrt(7)*Pi/2).

Equals Sum_{n>=0} (-1)^n * (2*n+1) / (n*(n+1)/2 + 1).

A258408 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^(2*k)) dx.

Original entry on oeis.org

5, 7, 7, 3, 3, 2, 1, 2, 0, 1, 8, 3, 9, 7, 9, 7, 0, 5, 5, 5, 2, 5, 4, 6, 9, 6, 2, 0, 1, 5, 9, 0, 4, 1, 5, 5, 0, 8, 0, 1, 1, 9, 3, 1, 3, 8, 3, 5, 6, 3, 4, 9, 2, 4, 5, 5, 8, 9, 0, 8, 8, 0, 3, 7, 5, 1, 5, 2, 5, 2, 1, 6, 4, 5, 1, 9, 8, 7, 7, 8, 1, 3, 5, 0, 6, 3, 7, 1, 0, 7, 0, 0, 0, 0, 0, 7, 1, 5, 4, 0, 9, 7, 8, 4, 7, 8

Offset: 0

Vaclav Kotesovec, May 29 2015

In general, Integral_{x=0..1} Product_{k>=1} (1-x^(m*k)) dx = Sum_{n} (-1)^n / (m*n*(3*n-1)/2 + 1) is equal to

if 0

(sqrt((24-m)*m) * (2*cosh(Pi/3*sqrt(24/m-1))-1))

if m = 24: Pi^2/(6*sqrt(3)) = A258414

if m > 24: 8*sqrt(3)*Pi*sin(Pi/6*sqrt(1-24/m)) /

(sqrt((m-24)*m) * (2*cos(Pi/3*sqrt(1-24/m))-1)).

Integral_{x=0..1} Product_{k=1..n} (1+x^(m*k)) dx, where m >= 1, is asymptotic to 2*(m+1)^(n+1)/(m*n^2).

Integral_{x=-1..1} Product_{k>=1} (1-x^(2*k)) dx = 8*Pi*sqrt(3/11) * sinh(sqrt(11)*Pi/6) / (2*cosh(sqrt(11)*Pi/3)-1) = 1.154664240367959678... . - Vaclav Kotesovec, Jun 02 2015

			0.5773321201839797055525469620159041550801193138356349245589088...

Vaclav Kotesovec, The integration of q-series

Cf. A258232, A258406, A258414.

evalf(4*Pi*sqrt(3/11) * sinh(sqrt(11)*Pi/6) / (2*cosh(sqrt(11)*Pi/3) - 1), 120);

RealDigits[4*Pi*Sqrt[3/11]*Sinh[Sqrt[11]*Pi/6] / (2*Cosh[Sqrt[11]*Pi/3] - 1),10,120][[1]]

Equals 4*Pi*sqrt(3/11) * sinh(sqrt(11)*Pi/6) / (2*cosh(sqrt(11)*Pi/3) - 1).

Showing 1-7 of 7 results.

cp's OEIS Frontend

A002107 Expansion of Product_{k>=1} (1 - x^k)^2.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Julia

Magma

Maple

Mathematica

PARI

PARI

PARI

Formula

A258232 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k) dx.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A258414 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^(24*k)) dx.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A258404 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^4 dx.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A258405 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^5 dx.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A258407 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^3 dx.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI