A059358 -id:A059358 - cp's OEIS frontend

A278945 Expansion of Sum_{k>=1} k(2k - 1)*x^k/(1 - x^k).

0, 1, 7, 16, 35, 46, 88, 92, 155, 169, 242, 232, 392, 326, 476, 496, 651, 562, 871, 704, 1050, 968, 1184, 1036, 1640, 1271, 1658, 1600, 2044, 1654, 2528, 1892, 2667, 2392, 2846, 2552, 3731, 2702, 3560, 3344, 4330, 3322, 4904, 3656, 5040, 4654, 5228, 4372, 6696, 4845, 6417, 5728, 7042, 5566, 8080, 6272, 8380, 7160, 8330, 6904, 10752

Offset: 0

Ilya Gutkovskiy, Dec 02 2016

Inverse Moebius transform of hexagonal numbers (A000384).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Mira Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
Mira Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric Weisstein's World of Mathematics, Hexagonal Number.
Index to sequences related to polygonal numbers.

Cf. A000203, A000384, A001157, A002117, A007437, A059358, A116913.

[0] cat [2*DivisorSigma(2, n) - DivisorSigma(1, n): n in [1..60]]; // Vincenzo Librandi, Dec 07 2016

nmax=60; CoefficientList[Series[Sum[k (2 k - 1) x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]
Flatten[{0, Table[2*DivisorSigma[2, n] - DivisorSigma[1, n], {n, 1, 100}]}] (* Vaclav Kotesovec, Dec 05 2016 *)

a(n) = if(n == 0, 0, my(f = factor(n)); 2 * sigma(f, 2) - sigma(f)); \\ Amiram Eldar, Dec 29 2024

G.f.: Sum_{k>=1} k*(2*k - 1)*x^k/(1 - x^k).
Dirichlet g.f.: (2*zeta(s-2) - zeta(s-1))*zeta(s).
a(n) = Sum_{d|n} d*(2*d - 1).
a(n) = 2*A001157(n) - A000203(n).
Sum_{k=1..n} a(k) ~ (2*zeta(3)/3) * n^3. - Amiram Eldar, Dec 29 2024

A320901 Expansion of Sum_{k>=1} x^k/(1 + x^k)^4.

Original entry on oeis.org

1, -3, 11, -23, 36, -49, 85, -143, 176, -188, 287, -433, 456, -479, 726, -959, 970, -1024, 1331, -1748, 1866, -1741, 2301, -3153, 2961, -2824, 3830, -4559, 4496, -4514, 5457, -6943, 6842, -6174, 7890, -9844, 9140, -8553, 11126, -13348, 12342, -11998, 14191, -16941

Offset: 1

Ilya Gutkovskiy, Oct 23 2018

sign

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

Cf. A000203, A000292, A000593, A001157, A001158, A002129, A050999, A051000, A059358, A138503, A320900, A321543.

Cf. A363598, A363616, A363617, A363631.

seq(coeff(series(add(x^k/(1+x^k)^4,k=1..n),x,n+1), x, n), n = 1 .. 45); # Muniru A Asiru, Oct 23 2018

nmax = 44; Rest[CoefficientList[Series[Sum[x^k/(1 + x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[(-1)^(d + 1) d (d + 1) (d + 2)/6, {d, Divisors[n]}], {n, 44}]

a(n) = sumdiv(n, d, (-1)^(d+1)*d*(d + 1)*(d + 2)/6); \\ Amiram Eldar, Jan 04 2025

G.f.: Sum_{k>=1} (-1)^(k+1)*A000292(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^(d+1)*d*(d + 1)*(d + 2)/6.
a(n) = (4*A000593(n) + 6*A050999(n) + 2*A051000(n) - 2*A000203(n) - 3*A001157(n) - A001158(n))/6.
a(n) = (A138503(n) + 3*A321543(n) + 2*A002129(n)) / 6. - Amiram Eldar, Jan 04 2025

A344992 a(n) = Sum_{1 <= i <= j <= k <= m <= n} gcd(i,j,k,m).

Original entry on oeis.org

1, 6, 18, 44, 83, 159, 249, 401, 592, 867, 1163, 1655, 2122, 2796, 3594, 4594, 5579, 7046, 8394, 10328, 12339, 14699, 17021, 20441, 23526, 27317, 31379, 36323, 40846, 47300, 52786, 59954, 67191, 75380, 83720, 94662, 103837, 115137, 126851, 141059, 153440

Offset: 1

Vaclav Kotesovec, Jun 05 2021

nonn

In general, if g.f.: 1/(1-x) * Sum_{j>=1} phi(j) * x^j/(1 - x^j)^k, where k > 2 and phi is the Euler totient function (A000010), then a(n) ~ zeta(k-1) * n^k / (k! * zeta(k)).

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Column k=4 of A345229.
Partial sums of A309323.
Cf. A000010, A059358, A272718, A309322, A344521.

Table[Sum[Sum[Sum[Sum[GCD[i, j, k, m], {i, 1, j}], {j, 1, k}], {k, 1, m}], {m, 1, n}], {n, 1, 100}]
nmax = 100; Rest[CoefficientList[Series[1/(1-x) * Sum[EulerPhi[k]*x^k/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]]
Accumulate[Table[Sum[EulerPhi[n/d] * d*(d+1)*(d+2)/6, {d, Divisors[n]}], {n, 1, 100}]] (* faster *)

a(n) = sum(i=1, n, sum(j=i, n, sum(k=j, n, sum(m=k, n, gcd([i, j, k, m]))))); \\ Michel Marcus, Jun 06 2021

G.f.: 1/(1-x) * Sum_{k>=1} phi(k) * x^k/(1 - x^k)^4, where phi is the Euler totient function (A000010).
a(n) = Sum_{k=1..n} Sum_{d|k} phi(k/d) * d*(d+1)*(d+2)/6.
a(n) ~ 15 * zeta(3) * n^4 / (4*Pi^4).
a(n) = Sum_{k=1..n} phi(k) * binomial(floor(n/k)+3,4). - Seiichi Manyama, Sep 13 2024

A363645 Expansion of Sum_{k>0} x^k/(1 - k*x^k)^4.

Original entry on oeis.org

1, 5, 11, 29, 36, 109, 85, 297, 256, 801, 287, 2881, 456, 5965, 3766, 17489, 970, 57385, 1331, 125681, 63498, 294933, 2301, 1072865, 24801, 1867009, 1087030, 4942561, 4496, 15697761, 5457, 28721057, 16895770, 63511593, 1404306, 225177013, 9140, 348661477

Offset: 1

Seiichi Manyama, Jun 13 2023

nonn

Cf. A167531, A363642.

Cf. A059358.

a[n_] := DivisorSum[n, (n/#)^(# - 1)*Binomial[# + 2, 3] &]; Array[a, 40] (* Amiram Eldar, Jul 18 2023 *)

a(n) = sumdiv(n, d, (n/d)^(d-1)*binomial(d+2, 3));

a(n) = Sum_{d|n} (n/d)^(d-1) * binomial(d+2,3).

A278947 Expansion of Sum_{k>=1} (k(5k - 3)/2)*x^k/(1 - x^k).

Original entry on oeis.org

0, 1, 8, 19, 42, 56, 107, 113, 190, 208, 298, 287, 483, 404, 589, 614, 806, 698, 1079, 875, 1302, 1202, 1471, 1289, 2035, 1581, 2062, 1990, 2541, 2060, 3142, 2357, 3318, 2978, 3544, 3178, 4641, 3368, 4435, 4166, 5390, 4142, 6106, 4559, 6279, 5798, 6517, 5453, 8339, 6042, 7998, 7142, 8778, 6944, 10070, 7822, 10445, 8930, 10390

Offset: 0

Ilya Gutkovskiy, Dec 02 2016

Inverse Moebius transform of heptagonal numbers (A000566).

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Mira Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
Mira Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric Weisstein's World of Mathematics, Heptagonal Number.
Index to sequences related to polygonal numbers.

Cf. A000203, A000566 (heptagonal numbers), A002117, A059358.

Inverse Moebius transforms of polygonal numbers: A007437 (k=3), A001157 (k=4), A116913 (k=5), A278945 (k=6), this sequence (k=7).

nmax=58; CoefficientList[Series[Sum[(k (5 k - 3)/2) x^k/(1 - x^k),  {k, 1, nmax}], {x, 0, nmax}], x]
Flatten[{0, Table[(5*DivisorSigma[2, n] - 3*DivisorSigma[1, n])/2, {n, 1, 100}]}] (* Vaclav Kotesovec, Dec 05 2016 *)

a(n) = if(n == 0, 0, my(f = factor(n)); (5 * sigma(f, 2) - 3 * sigma(f)) / 2); \\ Amiram Eldar, Dec 29 2024

G.f.: Sum_{k>=1} (k*(5*k - 3)/2)*x^k/(1 - x^k).
Dirichlet g.f.: (5*zeta(s-2) - 3*zeta(s-1))*zeta(s)/2.
a(n) = Sum_{d|n} d*(5*d - 3)/2.
a(n) = (5*A001157(n) - 3*A000203(n))/2.
Sum_{k=1..n} a(k) ~ (5*zeta(3)/6) * n^3. - Amiram Eldar, Dec 29 2024

A320941 Expansion of Sum_{k>=1} x^k*(1 + x^k)/(1 - x^k)^4.

Original entry on oeis.org

1, 6, 15, 36, 56, 111, 141, 240, 300, 446, 507, 791, 820, 1161, 1310, 1736, 1786, 2505, 2471, 3346, 3466, 4307, 4325, 5895, 5581, 7026, 7230, 8905, 8556, 11246, 10417, 13176, 13050, 15476, 15106, 19391, 17576, 21495, 21374, 25690, 23822, 30162, 27435, 33707, 32990, 37841, 35721

Offset: 1

Ilya Gutkovskiy, Oct 24 2018

Inverse Möbius transform of square pyramidal numbers (A000330).

Amiram Eldar, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms.

Cf. A000203, A000330, A001157, A001158, A007437, A059358.

Cf. A135539.

a:=series(add(x^k*(1+x^k)/(1-x^k)^4,k=1..100),x=0,48): seq(coeff(a,x,n),n=1..47); # Paolo P. Lava, Apr 02 2019

nmax = 47; Rest[CoefficientList[Series[Sum[x^k (1 + x^k)/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x]]
Table[Sum[d (d + 1) (2 d + 1)/6, {d, Divisors[n]}], {n, 47}]
Table[(DivisorSigma[1, n] + 3 DivisorSigma[2, n] + 2 DivisorSigma[3, n])/6, {n, 47}]

a(n) = my(f = factor(n)); (2*sigma(f, 3) + 3*sigma(f, 2) + sigma(f, 1)) / 6; \\ Amiram Eldar, Jan 03 2025

G.f.: Sum_{k>=1} A000330(k)*x^k/(1 - x^k).
a(n) = Sum_{d|n} d*(d + 1)*(2*d + 1)/6.
a(n) = (A000203(n) + 3*A001157(n) + 2*A001158(n))/6.
a(n) = Sum_{i=1..n} i^2*A135539(n,i). - Ridouane Oudra, Jul 22 2022
From Amiram Eldar, Jan 03 2025: (Start)
Dirichlet g.f.: zeta(s) * (2*zeta(s-3) + 3*zeta(s-2) + zeta(s-1)) / 6.
Sum_{k=1..n} a(k) ~ (zeta(4)/12) * n^4. (End)

A309730 Expansion of Sum_{k>=1} x^k * (1 - x^(3*k))/(1 - x^k)^4.

Original entry on oeis.org

1, 5, 11, 24, 32, 61, 65, 109, 120, 172, 167, 279, 236, 343, 358, 470, 410, 630, 515, 762, 706, 865, 761, 1193, 933, 1216, 1174, 1497, 1220, 1850, 1397, 1959, 1762, 2098, 1882, 2739, 2000, 2629, 2470, 3188, 2462, 3614, 2711, 3723, 3438, 3871, 3245, 4939, 3594, 4749, 4246, 5214

Offset: 1

Ilya Gutkovskiy, Aug 14 2019

Inverse Moebius transform of centered triangular numbers (A005448).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A001157, A002117, A005448, A007437, A059358.

nmax = 52; CoefficientList[Series[Sum[x^k (1 - x^(3 k))/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[3 (DivisorSigma[2, n] - DivisorSigma[1, n])/2 + DivisorSigma[0, n], {n, 1, 52}]

a(n)={sumdiv(n, d, 3*d*(d-1)/2 + 1)} \\ Andrew Howroyd, Aug 14 2019

a(n)={3*(sigma(n,2) - sigma(n))/2 + numdiv(n)} \\ Andrew Howroyd, Aug 14 2019

G.f.: Sum_{k>=1} (3*k*(k - 1)/2 + 1) * x^k/(1 - x^k).
a(n) = 3 * (sigma_2(n) - sigma_1(n))/2 + d(n).
From Amiram Eldar, Jan 02 2025: (Start)
Dirichlet g.f.: zeta(s) * (3 * zeta(s-2) - 3 * zeta(s-1) + 2 * zeta(s)) / 2.
Sum_{k=1..n} a(k) ~ (zeta(3)/2) * n^3. (End)

A321598 a(n) = Sum_{d|n} d*binomial(d+2,3).

Original entry on oeis.org

1, 9, 31, 89, 176, 375, 589, 1049, 1516, 2384, 3147, 4823, 5916, 8437, 10406, 14105, 16474, 22380, 25271, 33264, 37810, 47683, 52901, 68183, 73301, 91100, 100174, 122197, 130356, 161750, 169137, 205593, 219162, 259242, 272714, 330524, 338144, 400719, 421686, 493424

Offset: 1

Ilya Gutkovskiy, Nov 14 2018

Inverse Möbius transform of A002417.

Amiram Eldar, Table of n, a(n) for n = 1..10000
N. J. A. Sloane, Transforms.

Cf. A000292, A000335, A001157, A001158, A001159, A002417, A013663, A059358, A278403.

Table[Sum[d Binomial[d + 2, 3], {d, Divisors[n]}], {n, 40}]
nmax = 40; Rest[CoefficientList[Series[Sum[x^k (1 + 3 x^k)/(1 - x^k)^5, {k, 1, nmax}], {x, 0, nmax}], x]]
Table[(2 DivisorSigma[2, n] + 3 DivisorSigma[3, n] + DivisorSigma[4, n])/6, {n, 40}]

a(n) = my(f = factor(n)); (sigma(f, 4) + 3*sigma(f, 3) + 2*sigma(f, 2)) / 6; \\ Amiram Eldar, Jan 02 2025

G.f.: Sum_{k>=1} x^k*(1 + 3*x^k)/(1 - x^k)^5.
G.f.: Sum_{k>=1} k*A000292(k)*x^k/(1 - x^k).
L.g.f.: -log(Product_{k>=1} (1 - x^k)^A000292(k)) = Sum_{n>=1} a(n)*x^n/n.
Dirichlet g.f.: (zeta(s-4) + 3*zeta(s-3) + 2*zeta(s-2))*zeta(s)/6.
a(n) = (2*sigma_2(n) + 3*sigma_3(n) + sigma_4(n))/6.
a(n) = Sum_{d|n} A002417(d).
Sum_{k=1..n} a(k) ~ zeta(5) * n^5 / 30. - Vaclav Kotesovec, Feb 02 2019

A366933 Expansion of Sum_{k>=1} k^4 * x^k/(1 - x^k)^4.

Original entry on oeis.org

1, 20, 91, 340, 660, 1836, 2485, 5560, 7536, 13280, 14927, 31360, 29016, 49924, 60390, 89776, 84490, 152496, 131651, 226520, 227066, 299420, 282141, 514080, 415425, 581776, 614070, 850864, 711776, 1226520, 928977, 1442400, 1362042, 1693064, 1644930, 2609076

Offset: 1

Seiichi Manyama, Oct 29 2023

Cf. A064987, A366135, A366934.
Cf. A059358, A343544.
Cf. A343573.

a(n) = sumdiv(n, d, d^4*binomial(n/d+2, 3));

a(n) = Sum_{d|n} d^4 * binomial(n/d+2,3).

A366986 Square array T(n,k), n >= 1, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{d|n} binomial(d+k-1,k).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 4, 4, 3, 1, 5, 7, 7, 2, 1, 6, 11, 14, 6, 4, 1, 7, 16, 25, 16, 12, 2, 1, 8, 22, 41, 36, 31, 8, 4, 1, 9, 29, 63, 71, 71, 29, 15, 3, 1, 10, 37, 92, 127, 147, 85, 50, 13, 4, 1, 11, 46, 129, 211, 280, 211, 145, 52, 18, 2, 1, 12, 56, 175, 331, 498, 463, 371, 176, 74, 12, 6

Offset: 1

Seiichi Manyama, Oct 31 2023

			Square  array begins:
  1,  1,  1,  1,   1,   1,   1, ...
  2,  3,  4,  5,   6,   7,   8, ...
  2,  4,  7, 11,  16,  22,  29, ...
  3,  7, 14, 25,  41,  63,  92, ...
  2,  6, 16, 36,  71, 127, 211, ...
  4, 12, 31, 71, 147, 280, 498, ...
  2,  8, 29, 85, 211, 463, 925, ...

Columns k=0..5 give A000005, A000203, A007437, A059358, A073570, A101289.
T(n,n-1) gives A332508.
T(n,n) gives A343548.
Cf. A366977.

T(n, k) = sumdiv(n, d, binomial(d+k-1, k));

G.f. of column k: Sum_{j>=1} x^j/(1 - x^j)^(k+1).

cp's OEIS Frontend

A278945 Expansion of Sum_{k>=1} k*(2*k - 1)*x^k/(1 - x^k).

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A320901 Expansion of Sum_{k>=1} x^k/(1 + x^k)^4.

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A344992 a(n) = Sum_{1 <= i <= j <= k <= m <= n} gcd(i,j,k,m).

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A363645 Expansion of Sum_{k>0} x^k/(1 - k*x^k)^4.

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A278947 Expansion of Sum_{k>=1} (k*(5*k - 3)/2)*x^k/(1 - x^k).

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A320941 Expansion of Sum_{k>=1} x^k*(1 + x^k)/(1 - x^k)^4.

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A309730 Expansion of Sum_{k>=1} x^k * (1 - x^(3*k))/(1 - x^k)^4.

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A321598 a(n) = Sum_{d|n} d*binomial(d+2,3).

A278945 Expansion of Sum_{k>=1} k(2k - 1)*x^k/(1 - x^k).

A278947 Expansion of Sum_{k>=1} (k(5k - 3)/2)*x^k/(1 - x^k).