A258232 -id:A258232 - cp's OEIS frontend

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

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).

A319933 A(n, k) = [x^k] DedekindEta(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.

Original entry on oeis.org

1, 0, 1, 0, -1, 1, 0, -1, -2, 1, 0, 0, -1, -3, 1, 0, 0, 2, 0, -4, 1, 0, 1, 1, 5, 2, -5, 1, 0, 0, 2, 0, 8, 5, -6, 1, 0, 1, -2, 0, -5, 10, 9, -7, 1, 0, 0, 0, -7, -4, -15, 10, 14, -8, 1, 0, 0, -2, 0, -10, -6, -30, 7, 20, -9, 1, 0, 0, -2, 0, 8, -5, 0, -49, 0, 27, -10, 1

Offset: 0

Peter Luschny, Oct 02 2018

The columns are generated by polynomials whose coefficients constitute the triangle of signed D'Arcais numbers A078521 when multiplied with n!.

			[ 0] 1,   0,   0,    0,     0,    0,     0,     0,     0,     0, ... A000007
[ 1] 1,  -1,  -1,    0,     0,    1,     0,     1,     0,     0, ... A010815
[ 2] 1,  -2,  -1,    2,     1,    2,    -2,     0,    -2,    -2, ... A002107
[ 3] 1,  -3,   0,    5,     0,    0,    -7,     0,     0,     0, ... A010816
[ 4] 1,  -4,   2,    8,    -5,   -4,   -10,     8,     9,     0, ... A000727
[ 5] 1,  -5,   5,   10,   -15,   -6,    -5,    25,    15,   -20, ... A000728
[ 6] 1,  -6,   9,   10,   -30,    0,    11,    42,     0,   -70, ... A000729
[ 7] 1,  -7,  14,    7,   -49,   21,    35,    41,   -49,  -133, ... A000730
[ 8] 1,  -8,  20,    0,   -70,   64,    56,     0,  -125,  -160, ... A000731
[ 9] 1,  -9,  27,  -12,   -90,  135,    54,   -99,  -189,   -85, ... A010817
[10] 1, -10,  35,  -30,  -105,  238,     0,  -260,  -165,   140, ... A010818
    A001489,  v , A167541, v , A319931,  v ,         diagonal: A008705
           A080956       A319930      A319932

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. Fifth ed., Clarendon Press, Oxford, 2003.

M. Boylan, Exceptional congruences for the coefficients of certain eta-product newforms, J. Number Theory 98 (2003), no. 2, 377-389. MR1955423 (2003k:11071)
Steven R. Finch, Powers of Euler's q-Series, arXiv:math/0701251 [math.NT], 2007.
Vaclav 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.
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]
Tim Silverman, Counting Cliques in Finite Distant Graphs, arXiv preprint arXiv:1612.08085 [math.CO], 2016.
Michael Somos, Introduction to Ramanujan theta functions
Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Transpose of A286354.

Cf. A078521, A319574 (JacobiTheta3).

# DedekindEta is defined in A000594
for n in 0:10
    DedekindEta(10, n) |> println
end

DedekindEta := (x, n) -> mul(1-x^j, j=1..n):
A319933row := proc(n, len) series(DedekindEta(x, len)^n, x, len+1):
seq(coeff(%, x, j), j=0..len-1) end:
seq(print([n], A319933row(n, 10)), n=0..10);

eta[x_, n_] := Product[1 - x^j, {j, 1, n}];
A[n_, k_] := SeriesCoefficient[eta[x, k]^n, {x, 0, k}];
Table[A[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 10 2018 *)

from sage.modular.etaproducts import qexp_eta
def A319933row(n, len):
    return (qexp_eta(ZZ['q'], len+4)^n).list()[:len]
for n in (0..10):
    print(A319933row(n, 10))

A143440 Decimal expansion of the maximum value of the q-Pochhammer symbol along [ -1, 1].

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Aug 14 2008

			1.2283488670385751125...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, q-Pochhammer Symbol

Cf. A143441, A242168, A258232.

q0 = q /. FindRoot[QPochhammer'[q] == 0, {q, -1/2}, WorkingPrecision -> 300]; RealDigits[QPochhammer[q0], 10, 105] // First (* Jean-François Alcover, Dec 05 2013 *)

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).

A369880 Decimal expansion of sinh(Pi/2)/(Pi/2)^2.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Feb 04 2024

			0.93268131478635101777369755780799023506619209387697...

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. See p. 629, eq. (3.6); alternative link.

Cf. A000120, A019669, A060294, A096111, A091476, A185199, A258232, A308716, A367959.

RealDigits[Sinh[Pi/2]/(Pi/2)^2, 10, 120][[1]]

PARI
```
sinh(Pi/2)/(Pi/2)^2
```

Equals A060294 * A308716 = A308716 / A019669 = A185199 * A367959 = A367959 / A091476 = A367959 / A019669^2.

Equals Sum_{k>=0} (-1/16)^A000120(k)/D(k)^4, where D(k) = A096111(k-1) for k >= 1, and D(0) = 1 (Borwein and Borwein, 1992).

A258412 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^k dx.

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, May 29 2015

Integral_{x=0..1} Product_{k=1..n} (1+x^k)^k dx ~ 3*2^(n*(n+1)/2 + 1)/n^3.
Integral_{x=0..1} Product_{k=1..n} (1+x^k)   dx ~ 2^(n+2)/n^2.
Integral_{x=0..1} Product_{k>=1}   (1-x^k)   dx = A258232 = 0.3684125359314...
Integral_{x=0..1} Product_{k=1..n} (1-x^k)^n dx ~ 1/n.
Integral_{x=0..1} Product_{k=1..n} (1+x^k)^n dx ~ 2^(n^2 + 2)/n^3.

			0.298783365106567298770953772114...

Vaclav Kotesovec, The integration of q-series

Cf. A073592, A258232.

More digits from Vaclav Kotesovec, Oct 10 2023

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A258230 Denominator of Integral_{x=0..1} Product_{k=1..n} (1-x^k) 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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A319933 A(n, k) = [x^k] DedekindEta(x)^n, square array read by descending antidiagonals, A(n, k) for n >= 0 and k >= 0.

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A143440 Decimal expansion of the maximum value of the q-Pochhammer symbol along [ -1, 1].

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

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A369880 Decimal expansion of sinh(Pi/2)/(Pi/2)^2.

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A258412 Decimal expansion of Integral_{x=0..1} Product_{k>=1} (1-x^k)^k dx.

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