A002117 -id:A002117 - cp's OEIS frontend

A197070 Decimal expansion of the Dirichlet eta-function at 3.

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

Offset: 0

R. J. Mathar, Oct 09 2011

This constant is irrational by Apéry's theorem. - Charles R Greathouse IV, Feb 11 2024

			0.9015426773696957140498036211335874930737...

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
R. Barbieri, J. A. Mignaco and E. Remiddi, Electron form factors up to fourth order. I., Il Nuovo Cim. 11A (4) (1972) 824-864 Table II (4)
Su Hu, Min-soo Kim, Euler's integral, multiple cosine function and zeta values, arXiv:2201.011247 (2023), series last equation.
Seán Stewart, Problem 12206, The American Mathematical Monthly, Vol. 127, No. 8 (2020), p. 752.
Wikipedia, Dirichlet eta function.

Cf. A002117 (zeta(3)), A058312, A058313, A072691, A136675, A233090 (5*zeta(3)/8), A233091 (7*zeta(3)/8), A334582.

Maple
```
3*Zeta(3)/4 ; evalf(%) ;
```

RealDigits[3(Zeta[3])/4, 10, 75][[1]] (* Bruno Berselli, Dec 20 2011 *)

-polylog(3,-1) \\ Charles R Greathouse IV, Mar 28 2012

3/4*zeta(3) \\ Charles R Greathouse IV, Mar 28 2012

Equals 3*zeta(3)/4 = 3*A002117/4.
Also equals the integral over the unit cube [0,1]x[0,1]x[0,1] of 1/(1+x*y*z) dx dy dz. - Jean-François Alcover, Nov 24 2014
Equals Sum_{n>=1} (-1)^(n+1)/n^3. - Terry D. Grant, Aug 03 2016
Equals Lim_{n -> infinity} A136675(n)/A334582(n). - Petros Hadjicostas, May 07 2020
Equals Sum_{n>=1} AH(2*n)/n^2, where AH(n) = Sum_{k=1..n} (-1)^(k+1)/k = A058313(n)/A058312(n) is the n-th alternating harmonic number (Stewart, 2020). - Amiram Eldar, Oct 04 2021
Equals -int_0^1 log(x)log(1+x)/x dx [Barbieri] - R. J. Mathar, Jun 07 2024

A035528 Euler transform of A027656(n-1).

Original entry on oeis.org

0, 1, 1, 3, 3, 6, 9, 13, 19, 28, 42, 57, 84, 115, 164, 227, 313, 429, 588, 799, 1079, 1461, 1952, 2617, 3480, 4627, 6111, 8072, 10604, 13905, 18181, 23701, 30828, 39990, 51763, 66822, 86124, 110687, 142039, 181841, 232409, 296401, 377419, 479635, 608558, 770818

Offset: 0

Christian G. Bower

nonn

Also the weigh transform of A003602. - John Keith, Nov 17 2021

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.

Cf. A000219, A003602, A052847, A263150, A263352, A262876, A263136, A263141.

nmax = 50; CoefficientList[Series[-1 + Product[1/(1 - x^(2*k-1))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 19 2015 *)
nmax = 100; Flatten[{0, Rest[CoefficientList[Series[E^Sum[1/j*x^j/(1 - x^(2*j))^2, {j, 1, nmax}], {x, 0, nmax}], x]]}] (* Vaclav Kotesovec, Oct 10 2015 *)

a(n) ~ A^(1/2) * Zeta(3)^(11/72) * exp(-1/24 - Pi^4/(1728*Zeta(3)) + Pi^2 * n^(1/3)/(3*2^(8/3)*Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2^(4/3)) / (sqrt(3*Pi) * 2^(71/72) * n^(47/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Oct 02 2015

A233091 Decimal expansion of Sum_{i>=0} 1/(2*i+1)^3.

Original entry on oeis.org

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

Offset: 1

Bruno Berselli, Dec 04 2013

This constant is irrational. - Charles R Greathouse IV, Feb 03 2025

			1.0517997902646449997247708913225187419193630057979365215682376109241...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 42.

J. M. Borwein, I.J. Zucker, and J. Boersma, The evaluation of character Euler double sums, The Ramanujan Journal, April 2008, Volume 15, Issue 3, pp 377-405, see p. 17 c(3).
R. J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015, Table 22.

Cf. A002117: zeta(3); A197070: 3*zeta(3)/4; A233090: 5*zeta(3)/8.
Cf. A153071: sum( i >= 0, (-1)^i/(2*i+1)^3 ).
Cf. A251809: sum( i >= 0, (-1)^floor(i/2)/(2*i+1)^3 ).
Cf. A016755.

Mathematica
```
RealDigits[7 Zeta[3]/8, 10, 90][[1]]
```

7*zeta(3)/8 \\ Stefano Spezia, Oct 31 2024

Equals 7*zeta(3)/8.
Also equals -(1/16)*PolyGamma(2, 1/2). - Jean-François Alcover, Dec 18 2013
Equals Integral_{x=0..Pi/2} x * log(tan(x)) dx. - Amiram Eldar, Jun 29 2020
Equals Integral_{x=0..1} arcsin(x)*arccos(x)/x dx. - Amiram Eldar, Aug 03 2020

A258343 Expansion of Product_{k>=1} (1+x^k)^(k(k+1)(k+2)/6).

Original entry on oeis.org

1, 1, 4, 14, 36, 101, 260, 669, 1669, 4116, 9932, 23636, 55483, 128532, 294422, 667026, 1496232, 3324720, 7323570, 15998749, 34679966, 74622839, 159454379, 338472749, 713956569, 1496950669, 3120663129, 6469901522, 13343153563, 27379250529, 55907749171

Offset: 0

Vaclav Kotesovec, May 27 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000

Cf. A248882, A028377, A258341, A258342, A258344, A258345, A258346.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(binomial(i+2, 3), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, May 28 2018

nmax=40; CoefficientList[Series[Product[(1+x^k)^(k*(k+1)*(k+2)/6),{k,1,nmax}],{x,0,nmax}],x]

a(n) ~ (3*Zeta(5))^(1/10) / (2^(523/720) * 5^(2/5) * sqrt(Pi) * n^(3/5)) * exp(-2401 * Pi^16 / (10497600000000 * Zeta(5)^3) + 49*Pi^8 * Zeta(3) / (16200000 * Zeta(5)^2) - Zeta(3)^2 / (150*Zeta(5)) + (343*Pi^12 / (2430000000 * 2^(3/5) * 15^(1/5) * Zeta(5)^(11/5)) - 7*Pi^4 * Zeta(3) / (4500 * 2^(3/5) * 15^(1/5) * Zeta(5)^(6/5))) * n^(1/5) + (-49*Pi^8 / (1080000 * 2^(1/5) * 15^(2/5) * Zeta(5)^(7/5)) + Zeta(3) / (2^(6/5) * (15*Zeta(5))^(2/5))) * n^(2/5) + 7*Pi^4 / (180 * 2^(4/5) * (15*Zeta(5))^(3/5)) * n^(3/5) + 5*(15*Zeta(5))^(1/5) / 2^(12/5) * n^(4/5)), where Zeta(3) = A002117, Zeta(5) = A013663.

G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^4)). - Ilya Gutkovskiy, May 28 2018

A183097 a(n) = sum of powerful divisors d (including 1) of n.

Original entry on oeis.org

1, 1, 1, 5, 1, 1, 1, 13, 10, 1, 1, 5, 1, 1, 1, 29, 1, 10, 1, 5, 1, 1, 1, 13, 26, 1, 37, 5, 1, 1, 1, 61, 1, 1, 1, 50, 1, 1, 1, 13, 1, 1, 1, 5, 10, 1, 1, 29, 50, 26, 1, 5, 1, 37, 1, 13, 1, 1, 1, 5, 1, 1, 10, 125, 1, 1, 1, 5, 1, 1, 1, 130, 1, 1, 26, 5, 1, 1, 1, 29, 118, 1, 1, 5, 1, 1, 1, 13, 1, 10, 1, 5, 1, 1, 1, 61, 1, 50, 10, 130

Offset: 1

Jaroslav Krizek, Dec 25 2010

Sequence is not the same as A091051(n); a(72) = 130, A091051(72) = 58.

a(n) = sum of divisors d of n from set A001694 - powerful numbers.

			For n = 12, set of such divisors is {1, 4}; a(12) = 1+4 = 5.

Antti Karttunen, Table of n, a(n) for n = 1..16385
Index entries for sequences related to sums of divisors.

Cf. A001694, A091051, A183102.

Cf. A002117, A078434.

A183097 := proc(n)
    local a,pe,p,e ;
    a := 1;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if e > 1 then
            a := a* ( (p^(e+1)-1)/(p-1)-p) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Jun 02 2020

fun[p_,e_] := (p^(e+1)-1)/(p-1) - p; a[1] = 1; a[n_] := Times @@ (fun @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, May 14 2019 *)

A183097(n) = sumdiv(n, d, ispowerful(d)*d); \\ Antti Karttunen, Oct 07 2017

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(f[i,2]+1)-1) / (f[i,1]-1) - f[i,1]);} \\ Amiram Eldar, Dec 24 2023

a(n) = A000203(n) - A183098(n) = A183100(n) + 1.
a(1) = 1, a(p) = 1, a(pq) = 1, a(pq...z) = 1, a(p^k) = ((p^(k+1)-1) / (p-1))-p, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.
From Amiram Eldar, Dec 24 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * zeta(3*s-3) / zeta(6*s-6).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)^2/(3*zeta(3)) = 1.892451... . (End)

A073000 Decimal expansion of arctangent of 1/2.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Aug 03 2002

The angle at which you must shoot a cue ball on a standard pool table so that it will strike all four sides and return to its origin. [Barrow] - Robert G. Wilson v, Nov 29 2015

			Arctan(1/2)
=0.463647609000806116214256231461214402028537054286120263810933088720197864165... radians
=26°.56505117707798935157219372045329467120421429964522102798601631528806582148474...
=26°33'.9030706246793610943316232271976802722528579787132616791609789172839492890...
=26°33'54".184237480761665659897393631860816335171478722795700749658735037036957...
complement = 63°.43494882292201064842780627954670532879578570035477897201398368471...
supplement = 153°.4349488229220106484278062795467053287957857003547789720139836847...

John D. Barrow, One Hundred Essential Things You Didn't Know You Didn't Know, W. W. Norton & Co., NY & London, 2008.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 242.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Peter Bala, New series for old functions.
R. J. Mathar, Hierarchical Subdivision of the Simple Cubic Lattice, arXiv preprint arXiv:1309.3705 [math.MG], 2013.
Simon Plouffe, arctan(1/2).
Index entries for transcendental numbers

Cf. A001622, A002162, A002391, A105531, A254619.

evalf(arctan(0.5)) ; # R. J. Mathar, Aug 22 2013

Mathematica
```
RealDigits[ ArcTan[1/2], 10, 110] [[1]]
```

default(realprecision,2000); atan(1/2) \\ Anders Hellström, Nov 30 2015

Equals Pi/2 - A105199 = A019669-A105199. - R. J. Mathar, Aug 21 2013
From Peter Bala, Feb 04 2015: (Start)
Arctan(1/2) = 1/2*Sum_{k >= 0} (-1)^k/((2*k + 1)*4^k).
Define a pair of integer sequences A(n) = 4^n*(2*n + 1)!/n! and B(n) = A(n)*Sum_{k = 0..n} (-1)^k/((2*k + 1)*4^k). Both sequences satisfy the same second order recurrence equation u(n) = (12*n + 10)*u(n-1) + 16*(2*n - 1)^2*u(n-2). From this observation we obtain the continued fraction expansion 2*arctan(1/2) = 1 - 2/(24 + 16*3^2/(34 + 16*5^2/(46 + ... + 16*(2*n - 1)^2/((12*n + 10) + ...)))). See A002391, A105531 and A002162 for similar expansions.
Arctan(1/2) = 2/5 * Sum_{k >= 0} (4/5)^k/((2*k + 1)*binomial(2*k,k)).
Define a pair of integer sequences C(n) = 5^n*(2*n + 1)!/n! and D(n) = C(n)*Sum_{k = 0..n} (4/5)^k/((2*k + 1)*binomial(2*k,k)). Both sequences satisfy the same second order recurrence equation u(n) = (24*n + 10)*u(n-1) - 40*n*(2*n - 1)^2*u(n-2). From this observation we obtain the continued fraction expansion 5/2*arctan(1/2) = 1 + 4/(30 - 240/(58 - 600/(82 - ... - 40*n*(2*n - 1)/((24*n + 10) - ... )))).
Arctan(1/2) = 2/25 * Sum_{k >= 0} (24*k + 17)*(4/5)^(2*k)/( (4*k + 1)*(4*k + 3)*binomial(4*k,2*k) ).
Arctan(1/2) = 2/125 * Sum_{k >= 0} (1116*k^2 + 1446*k + 433)*(4/5)^(3*k)/( (6*k + 1)*(6*k + 3)*(6*k + 5)*binomial(6*k,3*k) ). (End)
Equals Integral_{x = 0..oo} exp(-2*x)*sin(x)/x dx. - Peter Bala, Nov 05 2019
Equals 2 * arccot(phi^3), where phi is the golden ratio (A001622). - Amiram Eldar, Jul 06 2023
Equals Sum_{n >= 1} i/(n*P(n, 2*i)*P(n-1, 2*i)) = (1/2)*Sum_{n >= 1} (-1)^(n+1)*4^n/(n*A098443(n)*A098443(n-1)), where i = sqrt(-1) and P(n, x) denotes the n-th Legendre polynomial. The n-th summand of the series is O( 1/(3 + 2*sqrt(2))^n ). - Peter Bala, Mar 16 2024

A051675 "Second factorials": Product_{k=1..n} k^(k^2).

Original entry on oeis.org

1, 16, 314928, 1352605460594688, 403107840000000000000000000000000, 4157825501361955044460594652554199040000000000000000000000000

Offset: 1

N. J. A. Sloane

Spyros G. Kanellos: Mathematical Researches and Results [in Greek]. Athens, 1965.

G. C. Greubel, Table of n, a(n) for n = 1..13
Karim Belabas and Henri Cohen, Numerical Algorithms for Number Theory Using PariGP, American Mathematical Society, 2021, p. 196.

Cf. A002109, A255321, A255323, A255344.

Cf. A243262.

[(&*[k^(k^2): k in [1..n]]): n in [1..10]]; // G. C. Greubel, Oct 14 2018

A051675 := proc(n) local k; mul(k^(k^2),k=1..n); end;

Table[Product[k^(k^2),{k,1,n}],{n,1,10}] (* Vaclav Kotesovec, Feb 20 2015 *)

for(n=1, 10, print1(prod(k=1,n, k^(k^2)), ", ")) \\ G. C. Greubel, Oct 14 2018

a(n) ~ n^(n*(n+1)*(2n+1)/6) / exp(n^3/9 - n/12 - Zeta(3)/(4*Pi^2)), where Zeta(3) = A002117 = 1.2020569031595942853997... . - Vaclav Kotesovec, Feb 20 2015

A161870 Convolution square of A000219.

Original entry on oeis.org

1, 2, 7, 18, 47, 110, 258, 568, 1237, 2600, 5380, 10870, 21652, 42350, 81778, 155676, 292964, 544846, 1003078, 1828128, 3301952, 5911740, 10499385, 18502582, 32371011, 56240816, 97073055, 166497412, 283870383, 481212656, 811287037, 1360575284, 2270274785, 3769835178, 6230705170, 10251665550, 16794445441

Offset: 0

Gary W. Adamson, Jun 20 2009

nonn

Equals [1,2,3,...] * [1,0,4,0,10,0,20,...] * [1,0,0,6,0,0,21,...] * [1,0,0,0,8,0,0,0,36,...] * ... - Gary W. Adamson, Jul 06 2009

Number of pairs of planar partitions of u and v where u + v = n. - Joerg Arndt, Apr 22 2014

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 19.
Paul Martin, Eric C. Rowell, and Fiona Torzewska, Classification of charge-conserving loop braid representations, arXiv:2301.13831 [math.QA], 2023.

Cf. A000219.

Column k=2 of A255961.

a:= proc(n) option remember; `if`(n=0, 1, 2*add(
      a(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 12 2015

nn = 36; CoefficientList[Series[Product[1/(1 - x^i)^(2 i), {i, 1, nn}] , {x, 0, nn}], x] (* Geoffrey Critzer, Nov 29 2014 *)

N=66;x='x+O('x^N); Vec(1/prod(k=1,N,(1-x^k)^k)^2) \\ Joerg Arndt, Apr 22 2014

G.f.: 1 / prod(k>=1, (1-x^k)^k )^2. - Joerg Arndt, Apr 22 2014
a(n) ~ Zeta(3)^(2/9) * exp(1/6 + 3*n^(2/3)*(Zeta(3)/2)^(1/3)) / (A^2 * 2^(1/18) * sqrt(3*Pi) * n^(13/18)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Feb 27 2015
G.f.: exp(2*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 29 2018

Added more terms, Joerg Arndt, Apr 22 2014

A183699 Decimal expansion of zeta(2)*zeta(3), the product of two Riemann zeta values.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Jan 06 2011

Equals the Dirichlet zeta-function Sum_{n>=1} A000203(n)/n^s at s=3.

			Equals 1.977304350297296118197085441485...

Cf. A000203, A013661, A002117, A347213.

Cf. A183700 (s=4).

Maple
```
evalf(Zeta(2)*Zeta(3));
```

RealDigits[Zeta[2] * Zeta[3], 10, 120][[1]] (* Amiram Eldar, Jun 12 2023 *)

zeta(2)*zeta(3) \\ Charles R Greathouse IV, Mar 04 2015

Equals A013661 * A002117.

A262811 Expansion of Product_{k>=1} 1/(1-x^(2k-1))^(2k-1).

Original entry on oeis.org

1, 1, 1, 4, 4, 9, 15, 22, 37, 56, 92, 133, 210, 310, 466, 696, 1013, 1495, 2160, 3141, 4495, 6462, 9172, 13024, 18387, 25840, 36213, 50500, 70280, 97302, 134522, 185105, 254245, 347938, 475036, 646676, 878145, 1189468, 1607095, 2166672, 2913794, 3910741

Offset: 0

Vaclav Kotesovec, Oct 03 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Cf. A000219, A003293, A035528, A161870, A262736, A292038.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(d::even, 0, d), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..45);  # Alois P. Heinz, Oct 05 2015

nmax = 60; CoefficientList[Series[Product[1/(1-x^(2*k-1))^(2*k-1), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(-1/12 + 3*Zeta(3)^(1/3)*n^(2/3)/2) * A * Zeta(3)^(5/36) / (2^(2/3) * sqrt(3*Pi) * n^(23/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

a(0) = 1, a(n) = (1/n)*Sum_{k=1..n} A050999(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 09 2017

cp's OEIS Frontend

A197070 Decimal expansion of the Dirichlet eta-function at 3.

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A035528 Euler transform of A027656(n-1).

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A233091 Decimal expansion of Sum_{i>=0} 1/(2*i+1)^3.

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A258343 Expansion of Product_{k>=1} (1+x^k)^(k*(k+1)*(k+2)/6).

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Maple

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Formula

A183097 a(n) = sum of powerful divisors d (including 1) of n.

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Maple

Mathematica

PARI

PARI

Formula

A073000 Decimal expansion of arctangent of 1/2.

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Maple

Mathematica

PARI

Formula

A051675 "Second factorials": Product_{k=1..n} k^(k^2).

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A258343 Expansion of Product_{k>=1} (1+x^k)^(k(k+1)(k+2)/6).

A262811 Expansion of Product_{k>=1} 1/(1-x^(2k-1))^(2k-1).