A002117 -id:A002117 - cp's OEIS frontend

A156616 G.f.: Product_{n>0} ((1+x^n)/(1-x^n))^n.

1, 2, 6, 16, 38, 88, 196, 420, 878, 1794, 3584, 7032, 13572, 25792, 48352, 89512, 163774, 296444, 531234, 943072, 1659560, 2896376, 5015700, 8622108, 14718652, 24960138, 42062200, 70458160, 117349856, 194381704, 320295312, 525123604

Offset: 0

R. J. Mathar, Feb 11 2009

Generating function for a sum over strict plane partitions weighted with 2 powered to their number of connected components.
The inverse Euler transform is apparently 2, 3, 6, 6, 10, 9, 14, 12, 18, 15, 22, 18, 26, 21, ..., A016825 interlaced with A008585. - R. J. Mathar, Apr 23 2009
In general, for m >= 1, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(m*k), then a(n) ~ exp(m/12 + 3/2 * (7*m*Zeta(3)/2)^(1/3) * n^(2/3)) * m^(1/6 + m/36) * (7*Zeta(3))^(1/6 + m/36) / (A^m * 2^(2/3 + m/9) * sqrt(3*Pi) * n^(2/3 + m/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 17 2015
In general, for m >= 0, if g.f. = Product_{k>=1} ((1+x^k)/(1-x^k))^(k^m), then a(n) ~ ((2^(m+2)-1) * Gamma(m+2) * Zeta(m+2) / (2^(2*m+3) * n))^((1-2*Zeta(-m))/(2*m+4)) * exp((m+2)/(m+1) * ((2^(m+2)-1) * n^(m+1) * Gamma(m+2) * Zeta(m+2) / 2^(m+1))^(1/(m+2)) + Zeta'(-m)) / sqrt((m+2)*Pi*n). - Vaclav Kotesovec, Aug 19 2015

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)
Ali H. Al-Saedi, Congruences for restricted plane overpartitions modulo 4 and 8, Raman. J. 48 (2) (2019) 251
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. 18.
Mirjana Vuletic, A generalization of MacMahon's formula, Trans. Am. Math. Soc. 361 (2009) 2789-2804.

Cf. A000219, A015128, A026007, A261384, A261386, A261389.
Cf. A206622, A206623, A206624, A261519, A261520.
Cf. A285462, A285447, A285460, A285461.
Cf. A285446, A285458, A285459.
Cf. A306081.

nmax = 40; CoefficientList[Series[Product[((1+x^k)/(1-x^k))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 17 2015 *)

{a(n)=polcoeff(exp(sum(m=1,n,(sigma(2*m,2)-sigma(m,2))/2*x^m/m)+x*O(x^n)),n)} \\ Paul D. Hanna, May 01 2010

Convolve A000219 with A026007.
O.g.f.: exp( Sum_{n>=1} (sigma_2(2n) - sigma_2(n))/2 *x^n/n ), where sigma_2(n) is the sum of squares of divisors of n (A001157). - Paul D. Hanna, May 01 2010
a(n) ~ exp(1/12 + 3 * 2^(-4/3) * (7*Zeta(3))^(1/3) * n^(2/3)) * (7*Zeta(3))^(7/36) / (A * 2^(7/9) * sqrt(3*Pi) * n^(25/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 17 2015
a(0) = 1, a(n) = (2/n)*Sum_{k=1..n} A076577(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 30 2017
G.f.: A(x) = exp( 2*Sum_{n >= 0} x^(2*n+1)/((2*n+1)*(1 - x^(2*n+1))^2) ). Cf. A000122 and A302237. - Peter Bala, Dec 23 2021

A000294 Expansion of g.f. Product_{k >= 1} (1 - x^k)^(-k*(k+1)/2).

Original entry on oeis.org

1, 1, 4, 10, 26, 59, 141, 310, 692, 1483, 3162, 6583, 13602, 27613, 55579, 110445, 217554, 424148, 820294, 1572647, 2992892, 5652954, 10605608, 19765082, 36609945, 67405569, 123412204, 224728451, 407119735, 733878402, 1316631730, 2351322765, 4180714647, 7401898452, 13051476707, 22922301583, 40105025130, 69909106888, 121427077241, 210179991927, 362583131144

Offset: 0

N. J. A. Sloane

Number of partitions of n if there are k(k+1)/2 kinds of k (k=1,2,...). E.g., a(3)=10 because we have six kinds of 3, three kinds of 2+1 because there are three kinds of 2 and 1+1+1+1. - Emeric Deutsch, Mar 23 2005
Euler transform of the triangular numbers 1,3,6,10,...
Equals A028377: [1, 1, 3, 9, 19, 46, 100, ...] convolved with the aerated version of A000294: [1, 0, 1, 0, 4, 0, 10, 0, 26, 0, 59, ...]. - Gary W. Adamson, Jun 13 2009
The formula for p3(n) in the article by S. Finch (page 2) is incomplete, terms with n^(1/2) and n^(1/4) are also needed. These terms are in the article by Mustonen and Rajesh (page 2) and agree with my results, but in both articles the multiplicative constant is marked only as C, resp. c3(m). The following is a closed form of this constant: Pi^(1/24) * exp(1/24 - Zeta(3) / (8*Pi^2) + 75*Zeta(3)^3 / (2*Pi^8)) / (A^(1/2) * 2^(157/96) * 15^(13/96)) = A255939 = 0.213595160470..., where A = A074962 is the Glaisher-Kinkelin constant and Zeta(3) = A002117. - Vaclav Kotesovec, Mar 11 2015 [The new version of "Integer Partitions" by S. Finch contains the missing terms, see pages 2 and 5. - Vaclav Kotesovec, May 12 2015]
Number of solid partitions of corner-hook volume n (see arXiv:2009.00592 among links for definition). E.g., a(2) = 1 because there is only one solid partition [[[2]]] with cohook volume 2; a(3) = 4 because we have three solid partitions with two 1's (entry at (1,1,1) contributes 1, another entry at (2,1,1) or (1,2,1) or (1,1,2) contributes 2 to corner-hook volume) and one solid partition with single entry 3 (which contributes 3 to the corner-hook volume). Namely as 3D arrays [[[1],[1]]],[[[1]],[[1]]],[[[1]],[[1]]], [[[3]]]. - Alimzhan Amanov, Jul 13 2021

R. Chandra, Tables of solid partitions, Proceedings of the Indian National Science Academy, 26 (1960), 134-139.
V. S. Nanda, Tables of solid partitions, Proceedings of the Indian National Science Academy, 19 (1953), 313-314.
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).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Alimzhan Amanov and Damir Yeliussizov, MacMahon's statistics on higher-dimensional partitions, arXiv:2009.00592 [math.CO], 2020. Mentions this sequence.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100.
A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100. [Annotated scanned copy]
R. Chandra, Tables of solid partitions, Proceedings of the Indian National Science Academy, 26 (1960), 134-139. [Annotated scanned copy]
Nicolas Destainville and Suresh Govindarajan, Estimating the asymptotics of solid partitions, J. Stat. Phys. 158 (2015) 950-967; arXiv:1406.5605 [cond-mat.stat-mech], 2014.
Steven Finch, Integer Partitions, September 22, 2004, page 2. [Cached copy, with permission of the author]
Vaclav Kotesovec, Graph - The asymptotic ratio
Ville Mustonen and R. Rajesh, Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer, J. Phys. A 36 (2003), no. 24, 6651-6659; arXiv:cond-mat/0303607 [cond-mat.stat-mech], 2003.
V. S. Nanda, Tables of solid partitions, Proceedings of the Indian National Science Academy, 19 (1953), 313-314. [Annotated scanned copy]
Damir Yeliussizov, Bounds on the number of higher-dimensional partitions, arXiv:2302.04799 [math.CO], 2023.

Cf. A000293, A007294, A007326, A255939, A028377.
Cf. also A000041, A000219, A000335, A000391, A000417, A000428, A255965.
Cf. also A278403 (log of o.g.f.).

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*(n+1)/2): seq(a(n), n=0..30);  # Alois P. Heinz, Sep 08 2008

a[0] = 1; a[n_] := a[n] = 1/(2*n)*Sum[(DivisorSigma[2, k]+DivisorSigma[3, k])*a[n-k], {k, 1, n}]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Mar 05 2014, after Vladeta Jovovic *)
nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(k*(k+1)/2),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Mar 11 2015 *)

a(n)=if(n<0, 0, polcoeff(exp(sum(k=1, n, x^k/(1-x^k)^3/k, x*O(x^n))), n)) \\ Joerg Arndt, Apr 16 2010

# uses[EulerTransform from A166861]
b = EulerTransform(lambda n: binomial(n+1, 2))
print([b(n) for n in range(37)]) # Peter Luschny, Nov 11 2020

a(n) = (1/(2*n))*Sum_{k=1..n} (sigma[2](k)+sigma[3](k))*a(n-k). - Vladeta Jovovic, Sep 17 2002
a(n) ~ Pi^(1/24) * exp(1/24 - Zeta(3) / (8*Pi^2) + 75*Zeta(3)^3 / (2*Pi^8) - 15^(5/4) * Zeta(3)^2 * n^(1/4) / (2^(7/4)*Pi^5) + 15^(1/2) * Zeta(3) * n^(1/2) / (2^(1/2)*Pi^2) + 2^(7/4) * Pi * n^(3/4) / (3*15^(1/4))) / (A^(1/2) * 2^(157/96) * 15^(13/96) * n^(61/96)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Mar 11 2015
G.f.: exp(Sum_{k>=1} (sigma_2(k) + sigma_3(k))*x^k/(2*k)). - Ilya Gutkovskiy, Aug 21 2018

More terms from Sascha Kurz, Aug 15 2002

A007408 Wolstenholme numbers: numerator of Sum_{k=1..n} 1/k^3.

Original entry on oeis.org

1, 9, 251, 2035, 256103, 28567, 9822481, 78708473, 19148110939, 19164113947, 25523438671457, 25535765062457, 56123375845866029, 56140429821090029, 56154295334575853, 449325761325072949, 2207911834254200646437, 245358578943756786493

Offset: 1

N. J. A. Sloane, Mira Bernstein

By Theorem 131 in Hardy and Wright, p^2 divides a(p - 1) for prime p > 5. - T. D. Noe, Sep 05 2002
p^3 divides a(p - 1) for prime p = 37. Primes p such that p divides a((p + 1)/2) are listed in A124787(n) = {3, 11, 17, 89}. - Alexander Adamchuk, Nov 07 2006
a(n)/A007409(n) is the partial sum towards zeta(3), where zeta(s) is the Riemann zeta function. - Alonso del Arte, Dec 30 2012
See the Wolfdieter Lang link under A103345 on Zeta(k, n) with the rationals for k=1..10, g.f.s and polygamma formulas. - Wolfdieter Lang, Dec 03 2013
Denominator of the harmonic mean of the first n cubes. - Colin Barker, Nov 13 2014

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford Univ. Press, 1971, page 104.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=1..200
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
D. Y. Savio, E. A. Lamagna and S.-M. Liu, Summation of harmonic numbers, pp. 12-20 of E. Kaltofen and S. M. Watt, editors, Computers and Mathematics, Springer-Verlag, NY, 1989.
M. D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications , J. Int. Seq. 13 (2010), 10.6.7, Section 4.3.2.

Cf. A001008, A007406, A007409 (denom), A002117, A124787, A249950.

A007408:=n->numer(sum(1/k^3,k=1..n)); map(%,[$1..20]); # M. F. Hasler, Nov 10 2006

Table[Numerator[Sum[1/k^3, {k, n}]], {n, 10}] (* Alonso del Arte, Dec 30 2012 *)
Table[Denominator[HarmonicMean[Range[n]^3]],{n,20}] (* Harvey P. Dale, Aug 20 2017 *)
Accumulate[1/Range[20]^3]//Numerator (* Harvey P. Dale, Aug 28 2023 *)

a(n)=numerator(sum(k=1,n,1/k^3)) \\ Charles R Greathouse IV, Jul 19 2011

from fractions import Fraction
from itertools import accumulate, count, islice
def A007408gen(): yield from map(lambda x: x.numerator, accumulate(Fraction(1, k**3) for k in count(1)))
print(list(islice(A007408gen(), 20))) # Michael S. Branicky, Jun 26 2022

Sum_{k = 1 .. n} 1/k^3 = sqrt(sum_{j = 1 .. n} sum_{i = 1 .. n} 1/(i * j)^3). - Alexander Adamchuk, Oct 26 2004

A005380 Expansion of 1 / Product_{k>=1} (1-x^k)^(k+1).

Original entry on oeis.org

1, 2, 6, 14, 33, 70, 149, 298, 591, 1132, 2139, 3948, 7199, 12894, 22836, 39894, 68982, 117948, 199852, 335426, 558429, 922112, 1511610, 2460208, 3977963, 6390942, 10206862, 16207444, 25596941, 40214896, 62868772, 97814358

Offset: 0

N. J. A. Sloane

Also, a(n) = number of partitions of the integer n where there are k+1 different kinds of part k for k = 1, 2, 3, ....
Also, a(n) = number of partitions of n objects of 2 colors. These are set partitions, the n objects are not labeled but colored, using two colors.  For each subset of size k there are k+1 different possibilities, i=0..k white and k-i black objects.
Also, a(n) = number of simple unlabeled graphs with n nodes of 2 colors whose components are complete graphs. - Geoffrey Critzer, Sep 27 2012

			We represent each summand, k, in a partition of n as k identical objects. Then we color each object. We have no regard for the order of the colored objects.
a(3) = 14 because we have:  www; wwb; wbb; bbb; ww + w; ww + b;  wb + w; wb + b; bb + w; bb + b; w + w + w; w + w + b; w + b + b; b + b + b, where the 2 colors are black b and white w. - _Geoffrey Critzer_, Sep 27 2012
a(3) = 14 because we have:  3; 3'; 3''; 3'''; 2 + 1; 2 + 1';  2' + 1; 2' + 1'; 2'' + 1; 2'' + 1'; 1 + 1 + 1; 1 + 1 + 1'; 1 + 1' + 1'; 1' + 1' + 1', where a part k of different sorts is given as k, k', k'', etc. - _Joerg Arndt_, Mar 09 2015
From _Alois P. Heinz_, Mar 09 2015: (Start)
The a(4) = 33 = 5 + 9 + 6 + 8 + 5 partitions of 4 objects of 2 colors are:
5 partitions for the integer partition of 4 = 1 + 1 + 1 + 1:
  01: {{b}, {b}, {b}, {b}}
  02: {{b}, {b}, {b}, {w}}
  03: {{b}, {b}, {w}, {w}}
  04: {{b}, {w}, {w}, {w}}
  05: {{w}, {w}, {w}, {w}}
9 partitions for the integer partition of 4 = 1 + 1 + 2:
  06: {{b}, {b}, {b,b}}
  07: {{b}, {w}, {b,b}}
  08: {{w}, {w}, {b,b}}
  09: {{b}, {b}, {w,b}}
  10: {{b}, {w}, {w,b}}
  11: {{w}, {w}, {w,b}}
  12: {{b}, {b}, {w,w}}
  13: {{b}, {w}, {w,w}}
  14: {{w}, {w}, {w,w}}
6 partitions for the integer partition of 4 = 2 + 2:
  15: {{b,b}, {b,b}}
  16: {{b,b}, {w,b}}
  17: {{b,b}, {w,w}}
  18: {{w,b}, {w,b}}
  19: {{w,b}, {w,w}}
  20: {{w,w}, {w,w}}
8 partitions for the integer partition of 4 = 1 + 3:
  21: {{b}, {b,b,b}}
  22: {{w}, {b,b,b}}
  23: {{b}, {w,b,b}}
  24: {{w}, {w,b,b}}
  25: {{b}, {w,w,b}}
  26: {{w}, {w,w,b}}
  27: {{b}, {w,w,w}}
  28: {{w}, {w,w,w}}
5 partitions for the integer partition of 4 = 4:
  29: {{b,b,b,b}}
  30: {{w,b,b,b}}
  31: {{w,w,b,b}}
  32: {{w,w,w,b}}
  33: {{w,w,w,w}}
Some see number partitions, others see set partitions, ...
(End)
It is obvious from the example of _Alois P. Heinz_ that a(n) enumerates multi-set partitions of a multi-set of n elements of two kinds. In the case that there is only one kind, this reduces to the usual case of numerical partitions. In the case that all the n elements are distinct, then this reduces to the case of set partitions. - _Michael Somos_, Mar 09 2015
There are a(3) = 14 plane partitions of 6 with trace 3; of 7 with trace 4; of 8 with trace 5; etc. See a formula above with the Stanley Exercise 7.99. - _Wolfdieter Lang_, Mar 09 2015
From _Daniel Forgues_, Mar 09 2015: (Start)
The a(3) = 14 = 4 + 6 + 4 partitions of 3 objects of 2 colors are:
4 partitions for the integer partition of 3 = 1 + 1 + 1:
  01: {{b}, {b}, {b}}
  02: {{b}, {b}, {w}}
  03: {{b}, {w}, {w}}
  04: {{w}, {w}, {w}}
6 partitions for the integer partition of 3 = 1 + 2:
  05: {{b}, {b,b}}
  06: {{w}, {b,b}}
  07: {{b}, {w,b}}
  08: {{w}, {w,b}}
  09: {{b}, {w,w}}
  10: {{w}, {w,w}}
4 partitions for the integer partition of 3 = 3:
  11: {{b,b,b}}
  12: {{w,b,b}}
  13: {{w,w,b}}
  14: {{w,w,w}}
The a(2) = 6 = 3 + 3 partitions of 2 objects of 2 colors are:
3 partitions for the integer partition of 2 = 1 + 1:
  01: {{b}, {b}}
  02: {{b}, {w}}
  03: {{w}, {w}}
3 partitions for the integer partition of 2 = 2:
  04: {{b,b}}
  05: {{w,b}}
  06: {{w,w}}
The a(1) = 2 partitions of 1 object of 2 colors are:
2 partitions for the integer partition of 1 = 1:
  01: {{b}}
  02: {{w}}
a(0) = 1: the empty partition, since empty sum is 0.
Triangle(sort of, since n_th row has p(n) = A000041 terms):
  1:  2
  2:  3, 3
  3:  4, 6, 4
  4:  5, 9, 6, 8, 5
  5:  6, ?, ?, ?, ?, ?, 6
  6:  7, ?, ?, ?, ?, ?, ?, ?, ?, ?, 7
Can we find a recurrence relation? (End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Exercise 7.99, p. 484 and pp. 548-549.

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
P. J. Cameron, Some sequences of integers, Discrete Math., 75 (1989), 89-102; also in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
Carlos A. A. Florentino, Plethystic exponential calculus and permutation polynomials, arXiv:2105.13049 [math.CO], 2021. Mentions this sequence.
Vaclav Kotesovec, Graph - The asymptotic ratio
P. A. MacMahon, Memoir on symmetric functions of the roots of systems of equations, Phil. Trans. Royal Soc. London, 181 (1890), 481-536; Coll. Papers II, 32-87.
N. J. A. Sloane, Transforms
R. P. Stanley, Theory and Applications of Plane Partitions: Part 2, Studies in Appl. Math., 1 (1971), 259-279.
R. P. Stanley, The conjugate trace and trace of a plane partition, J. Combin. Theory, vol. A14 53-65 1973, esp. p. 64.

Row sums of A054225.
Column k=2 of A075196.
Cf. A000219, A219555, A217093, A255050, A255052, A089353, A298988.

mul( (1-x^i)^(-i-1),i=1..80); series(%,x,80); seriestolist(%);
# second Maple program:
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..40); # Alois P. Heinz, Sep 08 2008

max = 31; f[x_] = Product[ 1/(1-x^k)^(k+1), {k, 1, max}]; CoefficientList[ Series[ f[x], {x, 0, max}], x] (* Jean-François Alcover, Nov 08 2011, after g.f. *)
etr[p_] := Module[{b}, b[n_] := b[n] = Module[{d, j}, If[n==0, 1, Sum[ Sum[ d*p[d], {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]]; b]; a = etr[#+1&]; Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Nov 23 2015, after Alois P. Heinz *)

a(n)=polcoeff(prod(i=1,n,(1-x^i+x*O(x^n))^-(i+1)),n)

EULER transform of b(n) = n+1.
a(n) ~ Zeta(3)^(13/36) * 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 * 2^(23/36) * 3^(1/2) * Pi * n^(31/36)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Mar 07 2015
a(n) = A089353(n+m, m), n >= 1, for each m >= n. a(0) =1. See the Stanley reference, Exercise 7.99. - Wolfdieter Lang, Mar 09 2015
G.f.: exp(Sum_{k>=1} (sigma_1(k) + sigma_2(k))*x^k/k). - Ilya Gutkovskiy, Aug 11 2018

Edited by Christian G. Bower, Sep 07 2002
New name from Joerg Arndt, Mar 09 2015
Restored 1995 name. - N. J. A. Sloane, Mar 09 2015

A264541 a(n) = numerator(Jtilde3(n)).

Original entry on oeis.org

0, 1, 65, 13247, 704707, 660278641, 357852111131, 309349386395887, 240498440880062263, 148443546307725010253, 61760947097005048531, 13658972396318235617977, 723464275788899734058353751, 489812222050789870424202126629, 2614176630672654770175367214389, 204702102697072009862200307064701369

Offset: 0

Michel Marcus, Nov 17 2015

Jtilde3(n) are Apéry-like rational numbers that arise in the calculation of zetaQ(3), the spectral zeta function for the non-commutative harmonic oscillator using a Gaussian hypergeometric function.

G. C. Greubel, Table of n, a(n) for n = 0..345
Takashi Ichinose, Masato Wakayama, Special values of the spectral zeta function of the non-commutative harmonic oscillator and confluent Heun equations, Kyushu Journal of Mathematics, Vol. 59 (2005) No. 1 p. 39-100.
Kazufumi Kimoto, Masato Wakayama, Apéry-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators, Kyushu Journal of Mathematics, Vol. 60 (2006) No. 2 p. 383-404 (see Table 2).

Cf. A002117 (zeta(3)), A260832 (Jtilde2), A264542 (denominators).

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692, A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

Numerator[Table[-2*Sum[(-1)^k*Binomial[-1/2, k]^2*Binomial[n, k]*Sum[ 1/(Binomial[-1/2, j]^2*(2*j + 1)^3), {j, 0, k - 1}], {k, 0, n}], {n, 0, 50}]] (* G. C. Greubel, Oct 24 2017 *)

a(n) = numerator(-2*sum(k=0, n, (-1)^k*binomial(-1/2, k)^2*binomial(n, k)*sum(j=0, k-1, 1/(binomial(-1/2,j)^2*(2*j+1)^3))));

Jtilde3(n) = J3(n) - J3(0)*Jtilde2(n) (normalization).

4n^2*J3(n) - (8n^2-8n+3)*J3(n-1) + 4(n-1)^2*J3(n-2) = 2^n*(n-1)!/(2n-1)!! with J3(0)=7*zeta(3) and J3(1)=21*zeta(3)/4 + 1/2.

A264542 a(n) = denominator(Jtilde3(n)).

Original entry on oeis.org

1, 2, 96, 17280, 860160, 774144000, 408748032000, 347163328512000, 266621436297216000, 163172319013896192000, 67488959156767948800, 14865958099336613068800, 785345441564243189248819200, 530893518497428395932201779200, 2831432098652951444971742822400, 221701133324526098141287462993920000

Offset: 0

Michel Marcus, Nov 17 2015

Jtilde3(n) are Apéry-like rational numbers that arise in the calculation of zetaQ(3), the spectral zeta function for the non-commutative harmonic oscillator using a Gaussian hypergeometric function.

G. C. Greubel, Table of n, a(n) for n = 0..345
Takashi Ichinose, Masato Wakayama, Special values of the spectral zeta function of the non-commutative harmonic oscillator and confluent Heun equations, Kyushu Journal of Mathematics, Vol. 59 (2005) No. 1 p. 39-100.
Kazufumi Kimoto, Masato Wakayama, Apéry-like numbers arising from special values of spectral zeta functions for non-commutative harmonic oscillators, Kyushu Journal of Mathematics, Vol. 60 (2006) No. 2 p. 383-404 (see Table 2).

Cf. A002117 (zeta(3)), A260832 (Jtilde2), A264541 (numerators).

Denominator[Table[-2*Sum[(-1)^k*Binomial[-1/2, k]^2*Binomial[n, k]* Sum[1/(Binomial[-1/2, j]^2*(2*j + 1)^3), {j, 0, k - 1}], {k, 0, n}], {n, 0, 50}]] (* G. C. Greubel, Oct 23 2017 *)

a(n) = denominator(-2*sum(k=0, n, (-1)^k*binomial(-1/2, k)^2*binomial(n, k)*sum(j=0, k-1, 1/(binomial(-1/2,j)^2*(2*j+1)^3))));

Jtilde3(n) = J3(n) - J3(0)*Jtilde2(n) (normalization).

4n^2*J3(n) - (8n^2-8n+3)*J3(n-1) + 4(n-1)^2*J3(n-2) = 2^n*(n-1)!/(2n-1)!! with J3(0)=7*zeta(3) and J3(1)=21*zeta(3)/4 + 1/2.

A028377 Expansion of Product_{m>0} (1+q^m)^(m(m+1)/2).

Original entry on oeis.org

1, 1, 3, 9, 19, 46, 100, 218, 460, 965, 1975, 3993, 7975, 15712, 30650, 59150, 113093, 214300, 402812, 751165, 1390714, 2557004, 4670770, 8479232, 15302657, 27462424, 49021252, 87057783, 153850769, 270614429, 473850031, 826125184, 1434286323, 2480145226

Offset: 0

N. J. A. Sloane

nonn

Convolved with aerated A000294: [1, 0, 2, 0, 4, 0, 10, 0, 26, ...] = A000294. - Gary W. Adamson, Jun 13 2009

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -n*(n+1)/2, g(n) = -1. - Seiichi Manyama, Nov 14 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A000294. - Gary W. Adamson, Jun 13 2009

Cf. A027999, A258341, A258342, A258343, A258344, A258345, A258346.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(binomial(i*(i+1)/2, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Aug 03 2013

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[i*(i+1)/2, j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 13 2014, after Alois P. Heinz *)

a(n) ~ 7^(1/8) * exp(2 * 7^(1/4) * Pi * n^(3/4) / (3^(5/4) * 5^(1/4)) + 3^(3/2) * 5^(1/2) * Zeta(3) * n^(1/2) / (2 * 7^(1/2) * Pi^2) - 3^(13/4) * 5^(5/4) * Zeta(3)^2 * n^(1/4) / (4 * 7^(5/4) * Pi^5) + 2025 * Zeta(3)^3 / (98*Pi^8)) / (2^(49/24) * 15^(1/8) * n^(5/8)), where Zeta(3) = A002117. - Vaclav Kotesovec, Mar 11 2015
a(0) = 1 and a(n) = (1/(2*n)) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d^2*(d+1)*(-1)^(1+n/d). - Seiichi Manyama, Nov 14 2017
G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)^3)). - Ilya Gutkovskiy, May 28 2018

A112526 Characteristic function for powerful numbers.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0

Offset: 1

Franklin T. Adams-Watters, Sep 09 2005

A signed multiplicative variant is defined by b(n) = a(n)*mu(n) with mu = A008683, such that b(p^e)=0 if e=1 and b(p^e)= -1 if e>1. This has Dirichlet series Sum_{n>=1} b(n)/n = A005596 and Sum_{n>=1} b(n)/n^2 = A065471. - R. J. Mathar, Apr 04 2011

			a(72) = 1 because 72 = 2^3*3^2 has all exponents > 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Powerful Number.
Index entries for characteristic functions.

Differs from characteristic function of perfect powers A075802 at Achilles numbers A052486.
Cf. A001694 (powerful numbers), A124010, A001221, A027746.
Cf. A008683, A008966, A005596, A065471, A082695, A063524.
Cf. A002117, A078434, A005361.

a112526 1 = 1
a112526 n = fromEnum $ (> 1) $ minimum $ a124010_row n
-- Reinhard Zumkeller, Jun 03 2015, Sep 16 2011

cfpn[n_]:=If[n==1||Min[Transpose[FactorInteger[n]][[2]]]>1,1,0]; Array[ cfpn,120] (* Harvey P. Dale, Jul 17 2012 *)

for(n=1, 100, print1(direuler(p=2, n, (1+X^3)/(1-X^2))[n], ", ")) \\ Vaclav Kotesovec, Jul 15 2022

a(n) = ispowerful(n); \\ Amiram Eldar, Jul 02 2025

from sympy import factorint
def A112526(n): return int(all(e>1 for e in factorint(n).values())) # Chai Wah Wu, Sep 15 2024

Multiplicative with a(p^e) = 1 - 0^(e-1), e > 0 and p prime.
Dirichlet g.f.: zeta(2*s)*zeta(3*s)/zeta(6*s), e.g., A082695 at s=1.
a(n) * A008966(n) = A063524(n). - Reinhard Zumkeller, Sep 16 2011
a(n) = {m: Min{A124010(m,k): k=1..A001221(m)} > 1}. - Reinhard Zumkeller, Jun 03 2015
Sum_{k=1..n} a(k) ~ zeta(3/2)*sqrt(n)/zeta(3) + 6*zeta(2/3)*n^(1/3)/Pi^2. - Vaclav Kotesovec, Feb 08 2019
a(n) = Sum_{d|n} A005361(d)*A008683(n/d). - Ridouane Oudra, Jul 03 2025

A021002 Decimal expansion of zeta(2)zeta(3)zeta(4)*...

Original entry on oeis.org

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

Offset: 1

Andre Neumann Kauffman (ank(AT)nlink.com.br)

A very good approximation is 2e-Pi = ~2.29497100332829723225793155942... - Marco Matosic, Nov 16 2005

This constant is equal to the asymptotic mean of number of Abelian groups of order n (A000688). - Amiram Eldar, Oct 16 2020

			2.2948565916733137941835158313443112887131637994416686732758140300...

R. Ayoub, An Introduction to the Analytic Theory of Numbers, Amer. Math. Soc., 1963, p. 198-9.
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 5.1 Abelian group enumeration constants, p. 274.

Robert Price, Table of n, a(n) for n = 1..1000
Steven R. Finch, Abelian Group Enumeration Constants. [From the Wayback machine]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 86.
Felix Fontein and Pawel Wocjan, Quantum Algorithm for Computing the Period Lattice of an Infrastructure, arXiv preprint arXiv:1111.1348 [quant-ph], 2011.
Felix Fontein and Pawel Wocjan, On the probability of generating a lattice, Journal of Symbolic Computation, Vol. 64 (2014), pp. 3-15, arXiv preprint, arXiv:1211.6246 [math.CO], 2012-2013. - From _N. J. A. Sloane_, Jan 03 2013
Bernd C. Kellner, On asymptotic constants related to products of Bernoulli numbers and factorials, Integers, Vol. 9 (2009), Article #A08, pp. 83-106; alternative link; arXiv:0604505 [math.NT], 2006.
B. R. Srinivasan, On the Number of Abelian Groups of a Given Order, Acta Arithmetica, Vol. 23, No. 2 (1973), pp. 195-205, alternative link.
Eric Weisstein's World of Mathematics, Abelian Group.
Index entries for constants related to zeta

Cf. A068982 (reciprocal), A082868 (continued fraction).

Cf. A002117, A000688, A063966, A080729, A080730.

evalf(product(Zeta(n), n=2..infinity), 200);

p = Product[ N[ Zeta[n], 256], {n, 2, 1000}]; RealDigits[p, 10, 111][[1]] (* Robert G. Wilson v, Nov 22 2005 *)

prodinf(n=2,zeta(n)) \\ Charles R Greathouse IV, May 27 2015

Product of A080729 and A080730. - R. J. Mathar, Feb 16 2011

More terms from Simon Plouffe, Jan 07 2002
Further terms from Robert G. Wilson v, Nov 22 2005
Mathematica program fixed by Vaclav Kotesovec, Sep 20 2014

A013631 Continued fraction for zeta(3).

Original entry on oeis.org

1, 4, 1, 18, 1, 1, 1, 4, 1, 9, 9, 2, 1, 1, 1, 2, 7, 1, 1, 7, 11, 1, 1, 1, 3, 1, 6, 1, 30, 1, 4, 1, 1, 4, 1, 3, 1, 2, 7, 1, 3, 1, 2, 2, 1, 16, 1, 1, 3, 3, 1, 2, 2, 1, 6, 1, 1, 1, 6, 1, 1, 4, 428, 5, 1, 1, 3, 1, 1, 11, 2, 4, 4, 5, 4, 1, 5, 14, 1, 3, 1, 2, 19, 1, 2, 5, 1, 7, 1, 1, 1, 1, 1, 57, 3, 2, 14, 2

Offset: 0

N. J. A. Sloane, John Morrison (John.Morrison(AT)armltd.co.uk)

			zeta(3) = 1.2020569031595942... = 1 + 1/(4 + 1/(1 + 1/(18 + 1/(1 + ...)))). - _Harry J. Smith_, Apr 20 2009

Harry J. Smith, Table of n, a(n) for n = 0..19999
Eric Weisstein's World of Mathematics, Apery's Constant.
G. Xiao, Contfrac.
Index entries for continued fractions for constants.
Index entries for zeta function.

Cf. A002117 (decimal expansion), A078984, A078985 (convergents).

Cf. continued fractions for zeta(2)-zeta(20): A013679, A013680-A013696.

Mathematica
```
ContinuedFraction[ Zeta[3], 100]
```

{ allocatemem(932245000); default(realprecision, 21000); x=contfrac(zeta(3)); for (n=1, 20000, write("b013631.txt", n-1, " ", x[n])); } \\ Harry J. Smith, Apr 20 2009

Offset changed by Andrew Howroyd, Jul 10 2024

cp's OEIS Frontend

A156616 G.f.: Product_{n>0} ((1+x^n)/(1-x^n))^n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A000294 Expansion of g.f. Product_{k >= 1} (1 - x^k)^(-k*(k+1)/2).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

SageMath

Formula

Extensions

A007408 Wolstenholme numbers: numerator of Sum_{k=1..n} 1/k^3.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A005380 Expansion of 1 / Product_{k>=1} (1-x^k)^(k+1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A264541 a(n) = numerator(Jtilde3(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A264542 a(n) = denominator(Jtilde3(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A028377 Expansion of Product_{m>0} (1+q^m)^(m(m+1)/2).

Views

A021002 Decimal expansion of zeta(2)zeta(3)zeta(4)*...