A064027 -id:A064027 - cp's OEIS frontend

A224364 G.f.: exp( Sum_{n>=1} A064027(n)x^n/n ), where A064027(n) = (-1)^nSum_{d|n}(-1)^d*d^2.

1, 1, 2, 5, 10, 18, 32, 59, 106, 181, 305, 518, 867, 1418, 2301, 3724, 5966, 9448, 14862, 23263, 36165, 55802, 85609, 130732, 198574, 299941, 450946, 675153, 1006395, 1493598, 2207928, 3251926, 4771934, 6977018, 10166502, 14766512, 21379861, 30859013, 44409543, 63729443

Offset: 0

Paul D. Hanna, Apr 04 2013

nonn

"Number of the pyramid partition of pyramid partitions obtained from the fundamental pyramid partition after the removal of n bricks." [From the Bouttier et al. reference] - Joerg Arndt, Jul 03 2014

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 10*x^4 + 18*x^5 + 32*x^6 + 59*x^7 +...
where
log(A(x)) = x + 3*x^2/2 + 10*x^3/3 + 19*x^4/4 + 26*x^5/5 + 30*x^6/6 + 50*x^7/7 + 83*x^8/8 + 91*x^9/9 + 78*x^10/10 +...+ A064027(n)*x^n/n +...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 575.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Paul D. Hanna)
Jérémie Bouttier, Guillaume Chapuy, Sylvie Corteel, From Aztec diamonds to pyramids: steep tilings, arXiv:1407.0665 [math.CO], 2014-2017.
Balázs Szendrői, Non-commutative Donaldson-Thomas invariants and the conifold, Geometry & Topology 12.2 (2008): 1171-1202.
B. Young, Computing a pyramid partition generating function with dimer shuffling, arXiv:0709.3079 [math.CO], 2007-2008.
B. Young, Computing a pyramid partition generating function with dimer shuffling, Journal of Combinatorial Theory, Series A, Volume 116, Issue 2, February 2009, Pages 334-350.
B. Young, J. Bryan, Generating functions for colored 3D Young diagrams and the Donaldson-Thomas invariants of orbifolds, arXiv:0802.3948 [math.CO], 2008.

Cf. A064027.

nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k-1))^(2*k-1) / (1 - x^(2*k))^(2*k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 14 2017 *)

{a(n)=polcoeff(exp(sum(k=1, n, sumdiv(k, d, (-1)^d*d^2)*(-x)^k/k)+x*O(x^n)), n)}
for(n=0, 40, print1(a(n), ", "))

N=66; q='q+O('q^N); Vec(prod(k=1,N, (1+q^(2*k-1))^(2*k-1)/(1-q^(2*k))^(2*k) )) \\ Joerg Arndt, Jul 03 2014

G.f.: Product_{k>=1} (1+q^(2*k-1))^(2*k-1) / (1-q^(2*k))^(2*k). - Joerg Arndt, Jul 03 2014

a(n) ~ exp(1/6 + 3 * 2^(-5/3) * (7*Zeta(3))^(1/3) * n^(2/3)) * (7*Zeta(3))^(2/9) / (2^(25/36) * A^2 * sqrt(3*Pi) * n^(13/18)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 14 2017

A321543 a(n) = Sum_{d|n} (-1)^(d-1)*d^2.

Original entry on oeis.org

1, -3, 10, -19, 26, -30, 50, -83, 91, -78, 122, -190, 170, -150, 260, -339, 290, -273, 362, -494, 500, -366, 530, -830, 651, -510, 820, -950, 842, -780, 962, -1363, 1220, -870, 1300, -1729, 1370, -1086, 1700, -2158, 1682, -1500, 1850, -2318, 2366, -1590, 2210, -3390, 2451, -1953, 2900, -3230, 2810, -2460, 3172

Offset: 1

N. J. A. Sloane, Nov 23 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Apart from signs, same as A064027.
Glaisher's zeta_i, i=0..12: A048272, A002129, A321543, A138503, A279395, A321544, A321545, A321546, A321547, A321548, A321549, A321550, A321551.
Cf. A321552 - A321565, A321807 - A321836 for similar sequences.

with(numtheory):
a := n -> add( (-1)^(d-1)*d^2, d in divisors(n) ): seq(a(n), n = 1..40);
#  Peter Bala, Jan 11 2021

f[p_, e_] := (p^(2*e + 2) - 1)/(p^2 - 1); f[2, e_] := 2 - (2^(2*e + 2) - 1)/3; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 55] (* Amiram Eldar, Nov 04 2022 *)

apply( a(n)=sumdiv(n, d, (-1)^(d-1)*d^2), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} (-1)^(k-1)*k^2*x^k/(1 - x^k). - Ilya Gutkovskiy, Dec 23 2018
G.f.: Sum_{n >= 1} x^n*(1 - x^n)/(1 + x^n)^3. - Peter Bala, Jan 11 2021
Multiplicative with a(2^e) = 2 - (2^(2*e + 2) - 1)/3, and a(p^e) = (p^(2*e + 2) - 1)/(p^2 - 1) for p > 2. - Amiram Eldar, Nov 04 2022

A078306 a(n) = Sum_{d divides n} (-1)^(n/d+1)*d^2.

Original entry on oeis.org

1, 3, 10, 11, 26, 30, 50, 43, 91, 78, 122, 110, 170, 150, 260, 171, 290, 273, 362, 286, 500, 366, 530, 430, 651, 510, 820, 550, 842, 780, 962, 683, 1220, 870, 1300, 1001, 1370, 1086, 1700, 1118, 1682, 1500, 1850, 1342, 2366, 1590, 2210, 1710, 2451, 1953

Offset: 1

Vladeta Jovovic, Nov 22 2002

Seiichi Manyama, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Heekyoung Hahn, Convolution sums of some functions on divisors, arXiv:1507.04426 [math.NT], 2015.
Index entries for sequences mentioned by Glaisher

Cf. A000593, A064027, A026007, A001157.

Glaisher's zeta'_i (i=0..12): A048272, A000593, A078306, A078307, A284900, A284926, A284927, A321552, A321553, A321554, A321555, A321556, A321557

a[n_] := Sum[(-1)^(n/d+1)*d^2, {d, Divisors[n]}]; Array[a, 50] (* Jean-François Alcover, Apr 17 2014 *)
Table[CoefficientList[Series[-Log[Product[1/(x^k + 1)^k, {k, 1, 90}]], {x, 0, 80}], x][[n + 1]] n, {n, 1, 80}] (* Benedict W. J. Irwin, Jul 05 2016 *)

a(n) = sumdiv(n, d, (-1)^(n/d+1)*d^2); \\ Michel Marcus, Jul 06 2016

from sympy import divisors
print([sum((-1)**(n//d + 1)*d**2 for d in divisors(n)) for n in range(1, 51)]) # Indranil Ghosh, Apr 05 2017

G.f.: Sum_{n >= 1} n^2*x^n/(1+x^n).
Multiplicative with a(2^e) = (2*4^e+1)/3, a(p^e) = (p^(2*e+2)-1)/(p^2-1), p > 2.
L.g.f.: -log(Product_{ k>0 } 1/(x^k+1)^k) = Sum_{ n>0 } (a(n)/n)*x^n. - Benedict W. J. Irwin, Jul 05 2016
G.f.: Sum_{n >= 1} (-1)^(n+1) * x^n*(1 + x^n)/(1 - x^n)^3. - Peter Bala, Jan 14 2021
From Vaclav Kotesovec, Aug 07 2022: (Start)
Dirichlet g.f.: zeta(s) * zeta(s-2) * (1 - 2^(1-s)).
Sum_{k=1..n} a(k) ~ zeta(3) * n^3 / 4. (End)

A008457 a(n) = Sum_{ d >= 1, d divides n} (-1)^(n-d)*d^3.

Original entry on oeis.org

1, 7, 28, 71, 126, 196, 344, 583, 757, 882, 1332, 1988, 2198, 2408, 3528, 4679, 4914, 5299, 6860, 8946, 9632, 9324, 12168, 16324, 15751, 15386, 20440, 24424, 24390, 24696, 29792, 37447, 37296, 34398, 43344, 53747, 50654, 48020, 61544, 73458

Offset: 1

N. J. A. Sloane

The modular form (e(1)-e(2))(e(1)-e(3)) for GAMMA_0 (2) (with constant term -1/16 omitted).

a(n) = r_8(n)/16, where r_8(n) = A000143(n) is the number of integral solutions of Sum_{j=1..8} x_j^2 = n (with the order of the summands respected). See the Grosswald reference, and the Hardy reference, pp. 146-147, eq. (9.9.3) and sect. 9.10. - Wolfdieter Lang, Jan 09 2017

			G.f. = q + 7*q^2 + 28*q^3 + 71*q^4 + 126*q^5 + 196*q^6 + 344*q^7 + 583*q^8 + ...

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (31.6).
Emil Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121, eq. (9.19).
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.
F. Hirzebruch, T. Berger and R. Jung, Manifolds and Modular Forms, Vieweg, 1994, pp. 77, 133.
Hans Petersson, Modulfunktionen und Quadratische Formen, Springer-Verlag, 1982; p. 179.

G. C. Greubel, Table of n, a(n) for n = 1..5000
Meinhard Peters, Sums of nine squares, Acta Arith., Vol. 102 (2002), pp. 131-135.

Cf. A000143, A001158, A051000, A064027, A002129, A048272, A138503.

(1/16)*product((1+q^n)^8/(1-q^n)^8,n=1..60);

nmax = 40; Rest[CoefficientList[Series[Product[((1-(-q)^k)/(1+(-q)^k))^8, {k, 1, nmax}]/16, {q, 0, nmax}], q]] (* Vaclav Kotesovec, Sep 26 2015 *)
a[n_] := DivisorSum[n, (-1)^(n-#)*#^3&]; Array[a, 40] (* Jean-François Alcover, Dec 01 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x]^8 - 1) / 16, {x, 0, n}]; (* Michael Somos, Aug 10 2018 *)
f[2, e_] := (8^(e+1)-15)/7; f[p_, e_] := (p^(3*e+3)-1)/(p^3-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 21 2020 *)

{a(n) = if( n<1, 0, (-1)^n * sumdiv(n, d, (-1)^d * d^3))}; /* Michael Somos, Sep 25 2005 */

from math import prod
from sympy import factorint
def A008457(n): return prod((p**(3*(e+1))-(1 if p&1 else 15))//(p**3-1) for p, e in factorint(n).items()) # Chai Wah Wu, Jun 21 2024

Multiplicative with a(2^e) = (8^(e+1)-15)/7, a(p^e) = (p^(3*e+3)-1)/(p^3-1), p > 2. - Vladeta Jovovic, Sep 10 2001
a(n) = (-1)^n*(sum of cubes of even divisors of n - sum of cubes of odd divisors of n), see A051000. Sum_{n>0} n^3*x^n*(15*x^n-(-1)^n)/(1-x^(2*n)). - Vladeta Jovovic, Oct 24 2002
G.f.: Sum_{k>0} k^3 x^k/(1 - (-x)^k). - Michael Somos, Sep 25 2005
G.f.: (1/16)*(-1+(Product_{k>0} (1-(-q)^k)/(1+(-q)^k))^8). [corrected by Vaclav Kotesovec, Sep 26 2015]
Dirichlet g.f. zeta(s)*zeta(s-3)*(1-2^(1-s)+2^(4-2s)), Dirichlet convolution of A001158 and the quasi-finite (1,-2,0,16,0,0,...). - R. J. Mathar, Mar 04 2011
A138503(n) = -(-1)^n * a(n).
Bisection: a(2*k-1) = A001158(2*k-1), a(2*k) = 8*A001158(k) - A051000(k), k >= 1. In the Hardy reference a(n) = sigma^*3(n). - _Wolfdieter Lang, Jan 07 2017
G.f.: (theta_3(x)^8 - 1)/16, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 17 2018
Sum_{k=1..n} a(k) ~ Pi^4 * n^4 / 384. - Vaclav Kotesovec, Sep 21 2020

A279396 Triangle read by rows T(n, m) = sigma^(n-m)(m), n >= 1, m = 1, 2, ..., n, with sigma^(k)(n) given in a comment in A279395.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 3, 4, 1, 1, 7, 10, 5, 2, 1, 15, 28, 19, 6, 0, 1, 31, 82, 71, 26, 4, 2, 1, 63, 244, 271, 126, 30, 8, 2, 1, 127, 730, 1055, 626, 196, 50, 13, 3, 1, 255, 2188, 4159, 3126, 1230, 344, 83, 13, 0, 1, 511, 6562, 16511, 15626, 7564, 2402, 583, 91, 6, 2, 1, 1023, 19684, 65791, 78126, 45990, 16808, 4367, 757, 78, 12, 2

Offset: 1

Wolfdieter Lang, Jan 10 2017

The array A(k, n) = sigma^*A279395)%20=%20Sum">(k)(n) (notation of the Hardy reference, given also in a comment in A279395) = Sum{ d >= 1, d divides n} (-1)^(n-d)*d^k, for k >= 0 and n >=1, has the rows A112329, A113184, A064027, A008457, A279395, for k=0..4.
The triangle T(n, m) is obtained from the array A(k, n) read by upwards antidiagonals, with offset n=1.
The diagonals of triangle T are the rows of the array A. Each diagonal is multiplicative. See the given A-numbers above.
The row sums are given in A279397.
The column sums (with offset 0) are A000012, A000225, A034472, A099393, A034474, .. with o.g.f. G(m, z) = (-1)^m*Sum_{d  | m} (-1)^d/(1 - d*z), m >= 1.

			The triangle T(n, m) begins:
n\m 1   2    3    4    5    6   7  8  9 10
1:  1
2:  1   0
3:  1   1    2
4:  1   3    4    1
5:  1   7   10    5    2
6:  1  15   28   19    6    0
7:  1  31   82   71   26    4   2
8:  1  63  244  271  126   30   8  2
9:  1 127  730 1055  626  196  50 13  3
10: 1 255 2188 4159 3126 1230 344 83 13  0
...
n = 11: 1 511 6562 16511 15626 7564 2402 583 91 6 2,
n = 12: 1 1023 19684 65791 78126 45990 16808 4367 757 78 12 2.
n = 13: 1 2047 59050 262655 390626 277876 117650 33823 6643 882 122 20 2,
n = 14: 1 4095 177148 1049599 1953126 1673310 823544 266303 59293 9390 1332 190 14 0,
n = 15: 1 8191 531442 4196351 9765626 10058524 5764802 2113663 532171 96906 14642 1988 170 8 4.
...

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.

Cf. A279394, A112329, A113184, A064027, A008457, A279395, A000012, A000225, A034472, A099393, A034474.

T(n, m) = Sum_{ d >= 1, d divides m} (-1)^(m-d)*d^(n-m) = sigma^*_(n-m)(m), n >= 1, m = 1,2, ..., n. For the definition of
sigma^*_(k)(n) see the Hardy reference or a comment in A279395.
O.g.f triangle T: G(z, x) = Sum_{m>=0}
G(m, z)*(x*z)^m, with the column o.g.f. G( m, z) (with offset 0) given in a comment above.

A279395 a(n) = Sum_{ d >= 1, d divides n} (-1)^(n-d)*d^4.

Original entry on oeis.org

1, 15, 82, 271, 626, 1230, 2402, 4367, 6643, 9390, 14642, 22222, 28562, 36030, 51332, 69903, 83522, 99645, 130322, 169646, 196964, 219630, 279842, 358094, 391251, 428430, 538084, 650942, 707282, 769980, 923522, 1118479, 1200644, 1252830, 1503652, 1800253, 1874162, 1954830, 2342084, 2733742

Offset: 1

Wolfdieter Lang, Jan 09 2017

This is the k=4 member of the family sigma^*_k(n), defined in the Hardy reference, which is sigma_k(2*j+1) if n = 2*j+1 and sigma_k^e(2*j) - sigma_k^o(2*j) if n=2*j, where the superscript e and o stands for a restriction to even and odd divisors in the sum of their k-th powers, respectively.

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.

Amiram Eldar, Table of n, a(n) for n = 1..10000
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher

Cf. A112329 (k=0), A113184 (k=1), A064027 (k=2), A008457(k=3).

[&+[(-1)^(n-d)*d^4:d in Divisors(n)]:n in [1..40]]; // Marius A. Burtea, Aug 17 2019

# A version with signs - N. J. A. Sloane, Nov 23 2018
zet1:=(n,i)->add((-1)^(d-1)*d^i, d in divisors(n));
szet1:=i->[seq(zet1(n,i),n=1..120)];
szet1(4);

f[p_, e_] := If[p == 2, (2^(4*(e + 1)) - 31)/15, (p^(4*(e + 1)) - 1)/(p^4 - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 40] (* Amiram Eldar, Aug 17 2019 *)

a(n) = sumdiv(n, d, (-1)^(n-d)*d^4); \\ Michel Marcus, Jan 09 2017

a(n) = Sum_{ d >= 1, d divides n} (-1)^(n-d)*d^4.
Bisection: a(2*j-1) = A001159(2*j-1), a(2*j) = 16*A001159(j) - A051001(j), j >= 1. See the comment above for k=4, and the Hardy reference.
G.f.: Sum_{k>=1} k^4*x^k/(1-(-x)^k).
Multiplicative with a(2^k) = 2^4*(2^(4*k)-1)/(2^4-1) - 1 = (2^(4*(k+1)) - 31)/15 and a(p^k) = (p^(4*(k+1))-1)/(p^4-1) for primes p > 2 (see A001159).

A320900 Expansion of Sum_{k>=1} x^k/(1 + x^k)^3.

Original entry on oeis.org

1, -2, 7, -12, 16, -17, 29, -48, 52, -42, 67, -105, 92, -79, 142, -184, 154, -143, 191, -262, 266, -189, 277, -441, 341, -262, 430, -495, 436, -402, 497, -712, 634, -444, 674, -897, 704, -553, 878, -1118, 862, -766, 947, -1189, 1222, -807, 1129, -1753, 1254, -992

Offset: 1

Ilya Gutkovskiy, Oct 23 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A000203, A000217, A000593, A001157, A002129, A007437, A050999, A064027, A320901, A321543.

Cf. A363022, A363615, A363630.

seq(coeff(series(add(x^k/(1+x^k)^3,k=1..n),x,n+1), x, n), n = 1 .. 50); # Muniru A Asiru, Oct 23 2018

nmax = 50; Rest[CoefficientList[Series[Sum[x^k/(1 + x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[(-1)^(d + 1) d (d + 1)/2, {d, Divisors[n]}], {n, 50}]

a(n) = sumdiv(n, d, (-1)^(d+1)*d*(d + 1)/2); \\ Amiram Eldar, Jan 04 2025

G.f.: Sum_{k>=1} (-1)^(k+1)*A000217(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^(d+1)*d*(d + 1)/2.
a(n) = A000593(n) + A050999(n) - (A000203(n) + A001157(n))/2.
a(n) = (A002129(n) + A321543(n)) / 2. - Amiram Eldar, Jan 04 2025

A372625 Expansion of Sum_{k>=1} k^2 * x^(k^2) / (1 + x^k).

Original entry on oeis.org

1, -1, 1, 3, 1, -5, 1, 3, 10, -5, 1, -6, 1, -5, 10, 19, 1, -14, 1, -13, 10, -5, 1, 10, 26, -5, 10, -13, 1, -39, 1, 19, 10, -5, 26, 14, 1, -5, 10, -6, 1, -50, 1, -13, 35, -5, 1, 46, 50, -30, 10, -13, 1, -50, 26, -30, 10, -5, 1, -11, 1, -5, 59, 83, 26, -50, 1, -13, 10, -79

Offset: 1

Ilya Gutkovskiy, May 07 2024

sign

Cf. A064027, A066839, A078306, A095118, A321543, A321558, A348608.

nmax = 70; CoefficientList[Series[Sum[k^2 x^(k^2)/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DivisorSum[n, (-1)^(# + n/#) #^2 &, # <= Sqrt[n] &], {n, 1, 70}]

a(n) = Sum_{d|n, d <= sqrt(n)} (-1)^(d + n/d) * d^2.

A357555 a(n) is the numerator of Sum_{d|n} (-1)^(d+1) / d^2.

Original entry on oeis.org

1, 3, 10, 11, 26, 5, 50, 43, 91, 39, 122, 55, 170, 75, 52, 171, 290, 91, 362, 143, 500, 183, 530, 215, 651, 255, 820, 275, 842, 13, 962, 683, 1220, 435, 52, 1001, 1370, 543, 1700, 559, 1682, 125, 1850, 61, 2366, 795, 2210, 95, 2451, 1953, 2900, 935, 2810, 205, 3172

Offset: 1

Ilya Gutkovskiy, Oct 03 2022

			1, 3/4, 10/9, 11/16, 26/25, 5/6, 50/49, 43/64, 91/81, 39/50, 122/121, ...

Cf. A017667, A064027, A098987, A119682, A321543, A357556 (denominators).

Table[Sum[(-1)^(d + 1)/d^2, {d, Divisors[n]}], {n, 1, 55}] // Numerator
nmax = 55; CoefficientList[Series[Sum[(-1)^(k + 1) x^k/(k^2 (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] // Numerator // Rest

a(n) = numerator(sumdiv(n, d, (-1)^(d+1)/d^2)); \\ Michel Marcus, Oct 03 2022

from sympy import divisors
from fractions import Fraction
def a(n): return sum(Fraction((-1)**(d+1), d*d) for d in divisors(n, generator=True)).numerator
print([a(n) for n in range(1, 56)]) # Michael S. Branicky, Oct 03 2022

Numerators of coefficients in expansion of Sum_{k>=1} (-1)^(k+1) * x^k / (k^2 * (1 - x^k)).

A357556 a(n) is the denominator of Sum_{d|n} (-1)^(d+1) / d^2.

Original entry on oeis.org

1, 4, 9, 16, 25, 6, 49, 64, 81, 50, 121, 72, 169, 98, 45, 256, 289, 108, 361, 200, 441, 242, 529, 288, 625, 338, 729, 392, 841, 15, 961, 1024, 1089, 578, 49, 1296, 1369, 722, 1521, 800, 1681, 147, 1849, 88, 2025, 1058, 2209, 128, 2401, 2500, 2601, 1352, 2809, 243, 3025

Offset: 1

Ilya Gutkovskiy, Oct 03 2022

			1, 3/4, 10/9, 11/16, 26/25, 5/6, 50/49, 43/64, 91/81, 39/50, 122/121, ...

Cf. A017668, A064027, A098988, A321543, A334580, A357555 (numerators).

Table[Sum[(-1)^(d + 1)/d^2, {d, Divisors[n]}], {n, 1, 55}] // Denominator
nmax = 55; CoefficientList[Series[Sum[(-1)^(k + 1) x^k/(k^2 (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x] // Denominator // Rest

a(n) = denominator(sumdiv(n, d, (-1)^(d+1)/d^2)); \\ Michel Marcus, Oct 03 2022

from sympy import divisors
from fractions import Fraction
def a(n): return sum(Fraction((-1)**(d+1), d*d) for d in divisors(n, generator=True)).denominator
print([a(n) for n in range(1, 56)]) # Michael S. Branicky, Oct 03 2022

Denominators of coefficients in expansion of Sum_{k>=1} (-1)^(k+1) * x^k / (k^2 * (1 - x^k)).

cp's OEIS Frontend

A224364 G.f.: exp( Sum_{n>=1} A064027(n)*x^n/n ), where A064027(n) = (-1)^n*Sum_{d|n}(-1)^d*d^2.

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A321543 a(n) = Sum_{d|n} (-1)^(d-1)*d^2.

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A078306 a(n) = Sum_{d divides n} (-1)^(n/d+1)*d^2.

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A008457 a(n) = Sum_{ d >= 1, d divides n} (-1)^(n-d)*d^3.

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A279396 Triangle read by rows T(n, m) = sigma^*(n-m)(m), n >= 1, m = 1, 2, ..., n, with sigma^*(k)(n) given in a comment in A279395.

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A279395 a(n) = Sum_{ d >= 1, d divides n} (-1)^(n-d)*d^4.

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A320900 Expansion of Sum_{k>=1} x^k/(1 + x^k)^3.

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A224364 G.f.: exp( Sum_{n>=1} A064027(n)x^n/n ), where A064027(n) = (-1)^nSum_{d|n}(-1)^d*d^2.

A279396 Triangle read by rows T(n, m) = sigma^(n-m)(m), n >= 1, m = 1, 2, ..., n, with sigma^(k)(n) given in a comment in A279395.