A002117 -id:A002117 - cp's OEIS frontend

A052847 G.f.: 1 / Product_{k>=1} (1-x^k)^(k-1).

1, 0, 1, 2, 4, 6, 12, 18, 33, 52, 88, 138, 229, 354, 568, 880, 1378, 2110, 3260, 4942, 7527, 11320, 17031, 25394, 37842, 55956, 82630, 121300, 177677, 258980, 376626, 545352, 787784, 1133764, 1627657, 2329020, 3324559, 4731396, 6717774, 9512060

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Euler transform of sequence [0,1,2,3,...]. - Michael Somos, Jul 02 2004
Number of partitions of n objects of 2 colors, where each part must contain at least one of each color. - Franklin T. Adams-Watters, Jan 23 2006
Number of partitions of n without 1s, one kind of 2s, two kinds of 3s, etc. - Joerg Arndt, Jul 31 2011
From Vaclav Kotesovec, Oct 17 2015: (Start)
In general, if v>=0 and g.f. = Product_{k>=1} 1/(1-x^(k+v))^k, then a(n) ~ d1(v) * n^(v^2/6 - 25/36) * exp(-Pi^4 * v^2 / (432*Zeta(3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2^(2/3) - v * Pi^2 * n^(1/3) / (3 * 2^(4/3) * Zeta(3)^(1/3))) / (sqrt(3*Pi) * 2^(v^2/6 + 11/36) * Zeta(3)^(v^2/6 - 7/36)), where Zeta(3) = A002117.
d1(v) = exp(Integral_{x=0..infinity} (1/(x*exp((v-1)*x) * (exp(x)-1)^2) - (6*v^2-1) / (12*x*exp(x)) + v/x^2 - 1/x^3) dx).
d1(v) = (exp(Zeta'(-1) - v*Zeta'(0))) / Product_{j=0..v-1} j!, where Zeta'(0) = -A075700, Zeta'(-1) = A084448 and Product_{j=0..v-1} j! = A000178(v-1).
d1(v) = exp(1/12) * (2*Pi)^(v/2) / (A * G(v+1)), where A = A074962 is the Glaisher-Kinkelin constant and G is the Barnes G-function.
(End)

			1 + x^2 + 2*x^3 + 4*x^4 + 6*x^5 + 12*x^6 + 18*x^7 + 33*x^8 + 52*x^9 + ...
From _Gus Wiseman_, Jan 22 2019: (Start)
The partitions described in Franklin T. Adams-Watters's comment are (n = 2 through 6):
  {{12}}  {{112}}  {{1112}}    {{11112}}    {{111112}}
          {{122}}  {{1122}}    {{11122}}    {{111122}}
                   {{1222}}    {{11222}}    {{111222}}
                   {{12}{12}}  {{12222}}    {{112222}}
                               {{12}{112}}  {{122222}}
                               {{12}{122}}  {{112}{112}}
                                            {{112}{122}}
                                            {{12}{1112}}
                                            {{12}{1122}}
                                            {{12}{1222}}
                                            {{122}{122}}
                                            {{12}{12}{12}}
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 815
Vaclav Kotesovec, Graph - The asymptotic ratio

Cf. A005380, A000203, A001157, A052812.
Cf. A000219 (v=0), A052847 (v=1), A263358 (v=2), A263359 (v=3), A263360 (v=4), A263361 (v=5), A263362 (v=6), A263363 (v=7), A263364 (v=8).
Cf. A054974, A321407, A321760, A323654, A323655, A323656.

spec := [S,{B=Sequence(Z,1 <= card),C=Prod(B,B),S= Set(C)},unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:=etr(n-> n-1): seq(a(n), n=0..50); # Vaclav Kotesovec, Mar 04 2015 after Alois P. Heinz

Clear[a]; a[n_]:= a[n] = 1/n*Sum[(DivisorSigma[2,k]-DivisorSigma[1,k])*a[n-k],{k,1,n}]; a[0]=1; Table[a[n],{n,0,100}] (* Vaclav Kotesovec, Mar 04 2015 *)
nmax = 40; CoefficientList[Series[Product[1/(1-x^(k+1))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 16 2015 *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, (1 - x^k + x*O(x^n))^(k-1)), n))}

a(n) = 1/n*Sum_{k=1..n} (sigma[2](k)-sigma[1](k))*a(n-k).
G.f.: exp( Sum_{k>0} ( x^k / (1 - x^k) )^2 / k ).
G.f.: exp( sum(n>=0, (sigma[2](n)-sigma[1](n)) *x^n/n ) ). - Joerg Arndt, Jul 31 2011
a(n) ~ 2^(1/36) * Zeta(3)^(1/36) * exp(1/12 - Pi^4/(432*Zeta(3)) - Pi^2 * n^(1/3) / (3 * 2^(4/3) * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A * 3^(1/2) * n^(19/36)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 07 2015

Edited by Vladeta Jovovic, Sep 10 2002

A093602 Decimal expansion of Pi/sqrt(3) = sqrt(2*zeta(2)).

Original entry on oeis.org

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

Offset: 1

Lekraj Beedassy, May 14 2004

Volume of a cube with edge length 1 rotated about a space diagonal. See MathWorld Cube page. - Francis Wolinski, Mar 10 2019

Volume of a cone with unit radius and 60-degree opening angle, and so height sqrt(3). Equivalently, the volume of the cone formed by rotating a 30-60-90 degree triangle with unit short leg about the long leg. - Christoph B. Kassir, Sep 17 2022

			Pi/sqrt(3) = 1.8137993642342178505940782576421557322840662480927405755...

Harry J. Smith, Table of n, a(n) for n = 1..20000
Peter Bala, New series for old functions
Jean Dolbeault, Ari Laptev and Michael Loss, Lieb-Thirring inequalities with improved constants, Vol. 10, No. 4 (2008), pp. 1121-1126, preprint, arXiv:0708.1165 [math.AP], 2007.
Sandi Klavžar, James Tuite, and Ullas Chandran, The General Position Problem: A Survey, arXiv:2501.19385 [math.CO], 2025. See p. 4.
Eric Weisstein's World of Mathematics, No-Three-in-a-Line-Problem
Eric Weisstein's World of Mathematics, Cube
Index entries for transcendental numbers

Continued fraction expansion is A132116. - Jonathan Vos Post, Aug 10 2007
Equals twice A093766.
Cf. A343235 (using the reciprocal), A248682.

SetDefaultRealField(RealField(100)); R:= RealField(); Pi(R)/Sqrt(3); // G. C. Greubel, Mar 10 2019

RealDigits[Pi/Sqrt[3],10,120][[1]] (* Harvey P. Dale, Mar 04 2012 *)

default(realprecision, 20080); x=Pi*sqrt(3)/3; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b093602.txt", n, " ", d)); \\ Harry J. Smith, Jun 19 2009

numerical_approx(pi/sqrt(3), digits=100) # G. C. Greubel, Mar 10 2019

Equals Integral_{x=0..oo} x^(1/3)/(1+x^2) dx. - Jean-François Alcover, May 24 2013
Equals (3/2)*Integral_{x=0..oo} 1/(1+x+x^2) dx. - Bruno Berselli, Jul 23 2013
Equals Sum_{n >= 0} (1/(6*n+1) - 4/(6*n+2) - 5/(6*n+3) - 1/(6*n+4) + 4/(6*n+5) + 5/(6*n+6)). - Mats Granvik, Sep 23 2013
Equals (1/2) * Sum_{n >= 0} (14*n + 11)*(-1/3)^n/((4*n + 1)*(4*n + 3)*binomial(4*n,2*n)). For more series representations of this type see the Bala link. - Peter Bala, Feb 04 2015
From Peter Bala, Nov 02 2019: (Start)
Equals 3*Sum_{n >= 1} 1/( (3*n - 1)*(3*n - 2) ).
Equals 2 - 6*Sum_{n >= 1} 1/( (3*n - 1)*(3*n + 1)*(3*n + 2) ).
Equals 5!*Sum_{n >= 1} 1/( (3*n - 1)*(3*n - 2)*(3*n + 2)*(3*n + 4) ).
Equals 3*( 1 - 2*Sum_{n >= 1} 1/(9*n^2 - 1) ).
Equals 1 + Sum_{n >=1 } (-1)^(n+1)*(6*n + 1)/(n*(n + 1)*(3*n + 1)*(3*n - 2)).
Equals (27/2)*Sum_{n >= 1} (2*n + 1)/( (3*n - 1)*(3*n + 1)*(3*n + 2)*(3*n + 4) ).
Equals 3*Integral_{x = 0..1} 1/(1 + x + x^2) dx.
Equals 3*Integral_{x = 0..1} (1 + x)/(1 - x + x^2) dx.
Equals 3*Integral_{x = 0..oo} cosh(x)/cosh(3*x) dx. (End)
Equals Integral_{x = 0..oo} log(1+x^3)/x^3 dx. - Amiram Eldar, Aug 20 2020
Equals (27*S - 36)/24, where S = A248682. - Peter Luschny, Jul 22 2022
From Peter Bala, Nov 09 2023: (Start)
For any integer k, Pi/sqrt(3) = Sum_{n >= 0} (1/(n + k + 1/3) - 1/(n - k + 2/3)) = (1/3)*Sum_{n >= 0} (1/(n - k + 1/6) - 1/(n + k + 5/6)).
Equals (3/2)*Sum_{n >= 0} 1/((2*n + 1)*binomial(2*n, n)). (End)

A243262 Decimal expansion of the generalized Glaisher-Kinkelin constant A(2).

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 02 2014

Also known as the second Bendersky constant.

This is likely the same as the constant B considered in section 3 of the Choi and Srivastava link. - R. J. Mathar, Oct 03 2016

			1.03091675219739211419331309646694229...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15 Glaisher-Kinkelin constant, p. 137.

G. C. Greubel, Table of n, a(n) for n = 1..2002
J. Choi and H. M. Srivastava, Certain classes of series involving the zeta function, J. Math. Annal. Applic. 231 (1999) 91-117.
K. Kimoto, N. Kurokawa, C. Sonoki, M. Wakayama, Some examples of generalized zeta regularized products, Kodai Math. J. 27 (2004), 321-335.
Tobias Kyrion, A closed-form expression for zeta(3), arXiv:2008.05573 [math.GM], 2020.
Eric Weisstein's MathWorld, Glaisher-Kinkelin Constant

Cf. A051675, A055462, A240966, A255269.
Cf. A019727, A074962, A243263, A243264, A243265, A266553, A266554, A266555, A266556, A266557, A266558, A266559, A260662, A266560, A266562, A266563, A266564, A266565, A266566, A266567.
Cf. A002117 (zeta(3)).

RealDigits[Exp[Zeta[3]/(4*Pi^2)], 10, 99] // First
RealDigits[Exp[N[(BernoulliB[2]/4)*(Zeta[3]/Zeta[2]), 200]]]//First (* G. C. Greubel, Dec 31 2015 *)

exp(zeta(3)/(4*Pi^2)) \\ Felix Fröhlich, Jun 27 2019

A(k) = exp(B(k+1)/(k+1)*H(k)-zeta'(-k)), where B(k) is the k-th Bernoulli number and H(k) the k-th harmonic number.
A(0) = sqrt(2*Pi) (A019727),
A(1) = A = Glaisher-Kinkelin constant (A074962),
A(2) = exp(-zeta'(-2)) = exp(zeta(3)/(4*Pi^2)).
Equals exp(-A240966). - Vaclav Kotesovec, Feb 22 2015

A023871 Expansion of Product_{k>=1} (1 - x^k)^(-k^2).

Original entry on oeis.org

1, 1, 5, 14, 40, 101, 266, 649, 1593, 3765, 8813, 20168, 45649, 101591, 223654, 486046, 1045541, 2225167, 4692421, 9804734, 20318249, 41766843, 85218989, 172628766, 347338117, 694330731, 1379437080, 2724353422, 5350185097, 10449901555, 20304465729, 39254599832

Offset: 0

Olivier Gérard

In general, if g.f. = Product_{k>=1} 1/(1 - x^k)^(c2*k^2 + c1*k + c0) and c2 > 0, then a(n) ~ exp(4*Pi * c2^(1/4) * n^(3/4) / (3*15^(1/4)) + c1*Zeta(3) / Pi^2 * sqrt(15*n/c2) + (Pi * 5^(1/4) * c0 / (2*3^(3/4) * c2^(1/4)) - 15^(5/4) * c1^2 * Zeta(3)^2 / (2*c2^(5/4) * Pi^5)) * n^(1/4) + c1/12 + 75 * c1^3 * Zeta(3)^3 / (c2^2 * Pi^8) - 5*c0 * c1 * Zeta(3) / (4*c2 * Pi^2) - c2*Zeta(3) / (4*Pi^2)) * Pi^(c1/12) * (c2/15)^(1/8 + c0/8 + c1/48) / (A^c1 * 2^((c0 + 3)/2) * n^(5/8 + c0/8 + c1/48)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 08 2017
Let A(x) = Product_{k >= 1} (1 - x^k)^(-k^2). The sequence defined by u(n) := [x^n] A(x)^n is conjectured to satisfy the supercongruences u(n*p^r) == u(n*p^(r-1)) (mod p^(3*r)) for all primes p >= 7 and all positive integers n and r. See A380290. - Peter Bala, Feb 02 2025
a(n) is the number of partitions of n where there are k^2 sorts of part k. - Joerg Arndt, Feb 02 2025

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 1001 terms from Alois P. Heinz)
G. Almkvist, Asymptotic formulas and generalized Dedekind sums, Exper. Math., 7 (No. 4, 1998), pp. 343-359.
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. 21.

Euler transform of squares (A000290).
Cf. A000219, A023872-A023878, A294530, A380290.
Column k=2 of A144048. - Alois P. Heinz, Nov 02 2012

m:=40; R:=PowerSeriesRing(Rationals(), m); Coefficients(R! ( (&*[1/(1-x^k)^k^2: k in [1..m]]) )); // G. C. Greubel, Oct 29 2018

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1,
      add(add(d*d^2, d=divisors(j)) *a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..35); # Alois P. Heinz, Nov 02 2012

max = 31; Series[ Product[ 1/(1-x^k)^k^2, {k, 1, max}], {x, 0, max}] // CoefficientList[#, x]& (* Jean-François Alcover, Mar 05 2013 *)

m=40; x='x+O('x^m); Vec(prod(k=1, m, 1/(1-x^k)^k^2)) \\ G. C. Greubel, Oct 29 2018

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: n^2)
print([b(n) for n in range(32)]) # Peter Luschny, Nov 11 2020

a(n) = 1/n * Sum_{k=1..n} a(n-k)*sigma_3(k), n > 0, a(0)=1, where sigma_3(n) = A001158(n) = sum of cubes of divisors of n. - Vladeta Jovovic, Jan 20 2002
G.f.: Prod_{n>=1} exp(sigma_3(n)*x^n/n), where sigma_3(n) is the sum of cubes of divisors of n (=A001158(n)). - N-E. Fahssi, Mar 28 2010
G.f. (conjectured): 1/Product_{n>=1} E(x^n)^J2(n) where E(x) = Product_{n>=1} 1-x^n and J2(n) = A007434(n) [follows from the identity Sum_{d|n} J2(d) = n^2 - Peter Bala, Feb 02 2025]. - Joerg Arndt, Jan 25 2011
a(n) ~ exp(4 * Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) - Zeta(3) / (4*Pi^2)) / (2^(3/2) * 15^(1/8) * n^(5/8)), where Zeta(3) = A002117 = 1.2020569031595942853997... . - Vaclav Kotesovec, Feb 27 2015

Definition corrected by Franklin T. Adams-Watters and R. J. Mathar, Dec 04 2006

A088453 Decimal expansion of 1/zeta(3).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Sep 30 2003

This is the probability that three randomly chosen integers are relatively prime (see A018805). - Gary McGuire, Dec 13 2004
This is also the probability that a random integer is cubefree. - Eugene Salamin, Dec 13 2004
On the other hand, the probability that three randomly-chosen integers are pairwise relatively prime is given by A065473. - Charles R Greathouse IV, Nov 14 2011
This is also the 'probability' that a random algebraic number's denominator is equal to its leading coefficient, see Arno, Robinson, & Wheeler. - Charles R Greathouse IV, Nov 12 2014
This is the probability that a random point on a cubic lattice is visible from the origin, i.e., there is no other lattice point that lies on the line segment between this point and the origin. - Amiram Eldar, Jul 08 2020

			0.831907372580707468683126278821530734417...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6, p. 41.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 29.

Steven Arno, M. L. Robinson, and Ferell S. Wheeler, On denominators of algebraic numbers and integer polynomials, Journal of Number Theory 57:2 (April 1996), pp. 292-302.
Eric Weisstein's World of Mathematics, Relatively Prime.

Cf. A002117, A008683.

RealDigits[1/Zeta[3],10,120][[1]] (* Harvey P. Dale, May 31 2019 *)

fpprec : 200$ bfloat( 1/zeta(3))$ bfloat(%); /* Martin Ettl, Oct 15 2012 */

1/zeta(3) \\ Charles R Greathouse IV, Nov 12 2014

Equals 1/A002117.
From Amiram Eldar, Aug 20 2020: (Start)
Equals Sum_{k>=1} mu(k)/k^3, where mu is the Möbius function (A008683).
Equals Product_{p prime} (1 - 1/p^3). (End)

Entry revised by N. J. A. Sloane, Dec 16 2004

A255528 G.f.: Product_{k>=1} 1/(1+x^k)^k.

Original entry on oeis.org

1, -1, -1, -2, 1, 0, 4, 2, 8, -2, 4, -11, -1, -25, -5, -35, 13, -26, 49, -6, 110, 6, 159, -23, 182, -141, 129, -358, 62, -640, 39, -897, 237, -1013, 771, -914, 1793, -664, 3143, -565, 4635, -1157, 5727, -3119, 6121, -7041, 5642, -13088, 5097, -20758, 5879

Offset: 0

Vaclav Kotesovec, Feb 24 2015

sign

In general, if m >= 1 and g.f. = Product_{k>=1} 1/(1 + x^k)^(m*k), then a(n, m) ~ (-1)^n * exp(-m/12 + 3 * 2^(-5/3) * m^(1/3) * Zeta(3)^(1/3) * n^(2/3)) * 2^(m/18 - 5/6) * A^m * m^(1/6 - m/36) * Zeta(3)^(1/6 - m/36) * n^(m/36 - 2/3) / sqrt(3*Pi), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 13 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Vaclav Kotesovec)
Vaclav Kotesovec, Graph - the asymptotic ratio (250000 terms)
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. 20.

Cf. A000219, A026007, A026011, A073592, A262736.

Cf. A278710 (m=2), A279031 (m=3), A279411 (m=4), A279932 (m=5).

with(numtheory): A000219:=proc(n) option remember; if n = 0 then 1 else add(sigma[2](k)*A000219(n-k), k = 1..n)/n fi: end: A073592:=proc(n) option remember; if n = 0 then 1 else -add(sigma[2](k)*A073592(n-k), k = 1..n)/n fi: end: a:=proc(n); add(A073592(n-2*m)*A000219(m), m = 0..floor(n/2)): end: seq(a(n), n = 0..50); # Vaclav Kotesovec, Mar 09 2015

nmax=100; CoefficientList[Series[Product[1/(1+x^k)^k,{k,1,nmax}],{x,0,nmax}],x]

{a(n) = if(n<0, 0, polcoeff(exp(sum(k=1, n, (-1)^k * x^k / (1-x^k)^2 / k, x*O(x^n))), n))}
for(n=0, 100, print1(a(n), ", "))

a(n) ~ (-1)^n * A * Zeta(3)^(5/36) * exp(3*Zeta(3)^(1/3)*n^(2/3)/2^(5/3) - 1/12) / (2^(7/9) * sqrt(3*Pi) * n^(23/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Sep 29 2015

a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A078306(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 06 2017

A152648 Decimal expansion of 2*zeta(3).

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Dec 10 2008

A division by 2 is missing in Mezo's penultimate formula on page 4.

This constant is irrational but not known to be transcendental. - Charles R Greathouse IV, Sep 02 2024

			Equals 2.4041138063191885707994...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 43.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Ilham A. Aliev and Ayhan Dil, Tornheim-like series, harmonic numbers and zeta values, arXiv:2008.02488 [math.NT], 2020, p. 2.
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. (3).
David Borwein and J. M. Borwein, On an intriguing integral and some series related to zeta(4), Proc. Am. Math. Soc. 123 (1995) 1191-1198.
Istvan Mezo, Summation of Hyperharmonic Numbers, arXiv:0811.0042 [math.CO], 2008.
Michael Penn, a nice double sum., YouTube video, 2020.
Michael Penn, Euler's harmonic number identity, YouTube video, 2020.

Cf. A002117, A001008, A002805.

Cf. A060804 (continued fraction).

RealDigits[2*Zeta[3],10,120][[1]] (* Harvey P. Dale, Dec 02 2011 *)

default(realprecision, 20080); x=2*zeta(3); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b152648.txt", n, " ", d));  \\ Harry J. Smith, Jul 12 2009

Equals 2*A002117 = Sum_{j>=1} H(j)/j^2 where H(j) = A001008(j)/A002805(j).
Equals Integral_{x>=0} x^2/(exp(x)-1). - Jean-François Alcover, Nov 12 2013
Equals Sum_{m>=1} Sum_{n>=1} 1/(m*n*(m + n)). - Jean-François Alcover, Jun 17 2020
Equals Integral_{x=0..1} log(x)^2/(1-x) dx. - Amiram Eldar, Aug 03 2020
Equals the absolute value of psi''(1) = -2.404..., the 2nd derivative of the digamma function at 1. - R. J. Mathar, Aug 29 2023

A306633 Decimal expansion of zeta(2)/zeta(3).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 02 2019

Equals the asymptotic mean of the unitary abundancy index, lim_{n->oo} (1/n) * Sum{k=1..n} usigma(k)/k, where usigma(k) is the sum of the unitary divisors of k (A034448).
From Amiram Eldar, May 12 2023: (Start)
Equals the asymptotic mean of the abundancy index of the squarefree numbers (A005117).
In general, the asymptotic mean of the abundancy index of the k-free numbers (numbers that are not divisible by a k-th power other than 1) is zeta(2)/zeta(k+1) (Jakimczuk and Lalín, 2022). (End)

			1.3684327776202058757367658539847919411308139146524...

Rafael Jakimczuk and Matilde Lalín, Asymptotics of sums of divisor functions over sequences with restricted factorization structure, Notes on Number Theory and Discrete Mathematics, Vol. 28, No. 4 (2022), pp. 617-634, eq. (1).
V. Sitaramaiah and M. V. Subbarao, Some asymptotic formulae involving powers of arithmetic functions, Number Theory, Madras 1987, Springer, 1989, pp. 201-234, alternative link.

Cf. A000010, A001615, A002117, A005117, A013661 (asymptotic mean of sigma(k)/k), A034448, A065463, A253905, A322887.

RealDigits[Zeta[2]/Zeta[3],10, 100][[1]]

zeta(2)/zeta(3) \\ Michel Marcus, Mar 04 2019

Equals A013661/A002117 = 1/A253905.
Equals Sum_{k>=1} phi(k)/k^3, where phi is the Euler totient function (A000010). - Amiram Eldar, Jun 23 2020
Equals Product_{p prime} (1 + 1/(p*(p+1))). - Amiram Eldar, Aug 10 2020
Equals Sum_{k>=1} mu(k)^2/(k*psi(k)) (the sum of reciprocals of the squarefree numbers multiplied by their Dedekind psi function values, A001615). - Amiram Eldar, Aug 18 2020

A055462 Superduperfactorials: product of first n superfactorials.

Original entry on oeis.org

1, 1, 2, 24, 6912, 238878720, 5944066965504000, 745453331864786829312000000, 3769447945987085350501386572267520000000000, 6916686207999802072984424331678589933649915805696000000000000000

Offset: 0

Henry Bottomley, Jun 26 2000

Next term has 92 digits and is too large to display.

Starting with offset 1, a(n) is a 'Matryoshka doll' sequence with alpha=1, the multiplicative counterpart to the additive A000332. The sequence for m with alpha<=m<=L is then computed as Prod_{n=alpha..m}(Prod_{k=alpha..n}(Prod_{i=alpha..k}(i))). - Peter Luschny, Jul 14 2009

			a(4) = 1!2!3!4!*1!2!3!*1!2!*1! = 288*12*2*1 = 6912.

Miles Seventh, Table of n, a(n) for n = 0..20
Shyam Sunder Gupta, Fascinating Factorials, Exploring the Beauty of Fascinating Numbers, Springer (2025) Ch. 16, 411-442.

Cf. A000142, A000178, A002109, A036740, A255268, A255269, A306635.

Cf. A057527, A057528, A260404.

[n eq 0 select 1 else (&*[j^Binomial(n-j+2,2): j in [1..n]]): n in [0..10]]; // G. C. Greubel, Jan 31 2024

seq(mul(mul(mul(i, i=alpha..k), k=alpha..n), n=alpha..m), m=alpha..10); # Peter Luschny, Jul 14 2009

Table[Product[BarnesG[j], {j, k + 1}], {k, 10}] (* Jan Mangaldan, Mar 21 2013 *)
Table[Round[Exp[(n+2)*(n+3)*(2*n+5)/8] * Exp[PolyGamma[-3, n+3]] * BarnesG[n+3]^(n+3/2) / (Glaisher^(n+3) * (2*Pi)^((n+3)^2/4) * Gamma[n+3]^((n+2)^2/2))], {n, 0, 10}] (* Vaclav Kotesovec, Feb 20 2015 after Jan Mangaldan *)
Nest[FoldList[Times,#]&,Range[0,15]!,2]  (* Harvey P. Dale, Jul 14 2023 *)

a(n)=my(t=1);prod(k=2,n,t*=k!) \\ Charles R Greathouse IV, Jul 28 2011

[product(j^binomial(n-j+2,2) for j in range(1,n+1)) for n in range(11)] # G. C. Greubel, Jan 31 2024

a(n) = a(n-1)*A000178(n) = Product_{i=1..n} (i!)^(n-i+1) = Product_{i=1..n} i^((n-i+1)*(n-i+2)/2).
log a(n) = (1/6) n^3 log n - (11/36) n^3 + O(n^2 log n). - Charles R Greathouse IV, Jan 13 2012
a(n) = exp((6 + 13 n + 9 n^2 + 2 n^3 - 8*(n + 2)*log(A)-2*(n + 2)^2*log(2*Pi) + 4*(2 n + 1)*logG(n + 2) - 4*(n + 1)^2*logGamma(n + 2) + 8*psi(-3, n + 2))/8) where A is the Glaisher-Kinkelin constant, logG(z) is the logarithm of the Barnes G function (A000178), and psi(-3, z) is a polygamma function of negative order (i.e., the second iterated integral of logGamma(z)). - Jan Mangaldan, Mar 21 2013
a(n) ~ exp(Zeta(3)/(8*Pi^2) - (2*n+3)*(11*n^2 + 24*n - 3)/72) * n^((2*n+3)*(2*n^2 + 6*n + 3)/24) * (2*Pi)^((n+1)*(n+2)/4) / A^(n+3/2), where A = A074962 = 1.28242712910062263687... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.2020569031595942853997... . - Vaclav Kotesovec, Feb 20 2015

a(9) from N. J. A. Sloane, Dec 15 2008

A033431 a(n) = 2*n^3.

Original entry on oeis.org

0, 2, 16, 54, 128, 250, 432, 686, 1024, 1458, 2000, 2662, 3456, 4394, 5488, 6750, 8192, 9826, 11664, 13718, 16000, 18522, 21296, 24334, 27648, 31250, 35152, 39366, 43904, 48778, 54000, 59582, 65536, 71874, 78608, 85750, 93312, 101306, 109744, 118638, 128000, 137842

Offset: 0

Jeff Burch

Also the largest determinant of a 3 X 3 matrix with entries from {0..n}. - Jud McCranie, Aug 12 2001
4*a(n) is a perfect cube.
The positive terms comprise the principal diagonal of the convolution array A213821. - Clark Kimberling, Jul 04 2012
Volume of a pyramid (square base) with side n and height 6*n. - Wesley Ivan Hurt, Aug 25 2014

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Amelia Carolina Sparavigna, Generalized Sum of Stella Octangula Numbers, Politecnico di Torino (Italy, 2021).
Amelia Carolina Sparavigna, Cardano Formula and Some Figurate Numbers, Politecnico di Torino (Italy, 2021).
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000578, A002117, A002378, A033581, A117642, A213821.

[2*n^3: n in [0..30]]; // Vincenzo Librandi, Jun 26 2011

seq(2*n^3, n=0..39); # Nathaniel Johnston, Jun 26 2011

2 Range[0, 50]^3 (* Wesley Ivan Hurt, Aug 25 2014 *)

a(n)=2*n^3 \\ Charles R Greathouse IV, Sep 24 2015

G.f.: 2*x*(1 + 4*x + x^2) / (1 - x)^4. - R. J. Mathar, Feb 04 2011
a(n) = 2*A000578(n). - Omar E. Pol, May 14 2008
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Wesley Ivan Hurt, Aug 25 2014
a(n) = A002378(n)^2 - A002378(n^2). - Bruno Berselli, Oct 20 2016
E.g.f.: 2*x*(1 + 3*x + x^2)*exp(x). - G. C. Greubel, Jul 15 2017
From Amiram Eldar, Jan 10 2023: (Start)
Sum_{n>=1} 1/a(n) = zeta(3)/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = 3*zeta(3)/8. (End)

cp's OEIS Frontend

A052847 G.f.: 1 / Product_{k>=1} (1-x^k)^(k-1).

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A093602 Decimal expansion of Pi/sqrt(3) = sqrt(2*zeta(2)).

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A243262 Decimal expansion of the generalized Glaisher-Kinkelin constant A(2).

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A023871 Expansion of Product_{k>=1} (1 - x^k)^(-k^2).

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A088453 Decimal expansion of 1/zeta(3).

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A255528 G.f.: Product_{k>=1} 1/(1+x^k)^k.

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