A258405 -id:A258405 - cp's OEIS frontend

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

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)

A000728 Expansion of Product_{n>=1} (1-x^n)^5.

Original entry on oeis.org

1, -5, 5, 10, -15, -6, -5, 25, 15, -20, 9, -45, -5, 25, 20, 10, 15, 20, -50, -35, -30, 55, -50, 15, 80, 1, 50, -35, -45, -15, 5, -50, -25, -55, 85, 51, 50, 10, -40, 65, 10, -10, -115, 50, -115, -100, 85, 80, -30, 5, 20, 45, 70, 65, 45, -55, -100

Offset: 0

N. J. A. Sloane

sign

Newman, Morris; A table of the coefficients of the powers of eta(tau). Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216.
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
M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
M. Newman, A table of the coefficients of the powers of eta(tau), Nederl. Akad. Wetensch. Proc. Ser. A. 59 = Indag. Math. 18 (1956), 204-216. [Annotated scanned copy]
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Cf. A258405.

CoefficientList[QPochhammer[x]^5 + O[x]^60, x] (* Jean-François Alcover, Feb 10 2016 *)

a(0) = 1, a(n) = -(5/n)*Sum_{k=1..n} A000203(k)*a(n-k) for n > 0. - Seiichi Manyama, Mar 26 2017

G.f.: exp(-5*Sum_{k>=1} x^k/(k*(1 - x^k))). - Ilya Gutkovskiy, Feb 05 2018

A258406 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^2 dx.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 29 2015

			0.2538740823782760029885088938163329123847636343193313514756067...

Vaclav Kotesovec, The integration of q-series

Cf. A002107, A258232, A258407, A258404, A258405.

evalf(Sum(Sum(8*(n+1)*(-1)^n / ((n^2 - 2*k^2 + 2*k*n + n + 2) * (n^2 - 2*k^2 + 2*k*n + 5*n + 6)), k=0..n), n=0..infinity), 120);

RealDigits[NIntegrate[QPochhammer[x]^2, {x, 0, 1}, WorkingPrecision -> 120], 10, 106][[1]] (* Vaclav Kotesovec, Oct 10 2023 *)

Equals Sum_{n>=0} Sum_{k=0..n} 8*(n+1)*(-1)^n / ((n^2 - 2*k^2 + 2*k*n + n + 2) * (n^2 - 2*k^2 + 2*k*n + 5*n + 6)).

Equals Sum_{n>=0} Sum_{j=-floor(n/2)..floor(n/2)} (-1)^(n+j) / (n*(n+1)/2 - j*(3*j-1)/2 + 1).

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

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).

Showing 1-5 of 5 results.

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

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A000728 Expansion of Product_{n>=1} (1-x^n)^5.

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A258406 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^2 dx.

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

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

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