cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Previous Showing 11-20 of 32 results. Next

A363642 Expansion of Sum_{k>0} x^k/(1 - k*x^k)^3.

Original entry on oeis.org

1, 4, 7, 17, 16, 55, 29, 129, 100, 311, 67, 1135, 92, 1919, 1486, 5409, 154, 17038, 191, 33491, 20938, 67871, 277, 262861, 9701, 373127, 296110, 978727, 436, 3134821, 497, 5051969, 3898522, 10027655, 474146, 39352069, 704, 49808159, 48362926, 127403221, 862, 411286429, 947
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A363639, A363643, A363644.
Cf. A167531, A363645.
Cf. A007437, A363650.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(#-1) * Binomial[# + 1, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(d-1)*binomial(d+1, 2));

Formula

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

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

Views

Author

Ilya Gutkovskiy, Oct 23 2018

Keywords

Links

Crossrefs

Cf. A000203, A000217, A000593, A001157, A002129, A007437, A050999, A064027, A320901, A321543.
Cf. A363022, A363615, A363630.

Programs

  • Maple
    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
  • Mathematica
    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}]
  • PARI
    a(n) = sumdiv(n, d, (-1)^(d+1)*d*(d + 1)/2); \\ Amiram Eldar, Jan 04 2025

Formula

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

A363610 Expansion of Sum_{k>0} x^(3*k)/(1-x^k)^3.

Original entry on oeis.org

0, 0, 1, 3, 6, 11, 15, 24, 29, 42, 45, 69, 66, 93, 98, 129, 120, 175, 153, 216, 206, 255, 231, 343, 282, 366, 354, 447, 378, 550, 435, 594, 542, 648, 582, 828, 630, 819, 770, 978, 780, 1114, 861, 1161, 1072, 1221, 1035, 1529, 1143, 1494, 1346, 1644, 1326, 1878, 1482, 1953
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2023

Keywords

Links

Crossrefs

Cf. A000005, A000203, A001157, A002117, A065608, A363611.
Cf. A007437, A069153.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, Binomial[# - 1, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)
  • PARI
    my(N=60, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/(1-x^k)^3)))
  • PARI
    a(n) = my(f = factor(n)); (sigma(f, 2) - 3*sigma(f) + 2*numdiv(f)) / 2; \\ Amiram Eldar, Jan 01 2025

Formula

G.f.: Sum_{k>0} binomial(k-1,2) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d-1,2).
From Amiram Eldar, Jan 01 2025: (Start)
a(n) = (sigma_2(n) - 3*sigma_1(n) + 2*sigma_0(n)) / 2.
Dirichlet g.f.: zeta(s) * (zeta(s-2) - 3*zeta(s-1) + 2*zeta(s)) / 2.
Sum_{k=1..n} a(k) ~ (zeta(3)/6) * n^3. (End)

A364970 a(n) = Sum_{k=1..n} binomial(floor(n/k)+2,3).

Original entry on oeis.org

1, 5, 12, 26, 42, 73, 102, 152, 204, 278, 345, 464, 556, 693, 835, 1021, 1175, 1422, 1613, 1907, 2173, 2496, 2773, 3228, 3569, 4015, 4445, 4998, 5434, 6120, 6617, 7331, 7965, 8717, 9391, 10392, 11096, 12031, 12909, 14059, 14921, 16219, 17166, 18489, 19711, 21072, 22201
Offset: 1

Views

Author

Seiichi Manyama, Oct 23 2023

Keywords

Crossrefs

Partial sums of A007437.
Cf. A006218, A024916, A064602, A365409, A365439.

Programs

  • Mathematica
    Table[Sum[Binomial[Floor[n/k+2],3],{k,n}],{n,50}] (* Harvey P. Dale, Aug 04 2024 *)
  • PARI
    a(n) = sum(k=1, n, binomial(n\k+2, 3));
  • Python
    from math import isqrt
def A364970(n): return (-(s:=isqrt(n))**2*(s+1)*(s+2)+sum((q:=n//k)*(3*k*(k+1)+(q+1)*(q+2)) for k in range(1,s+1)))//6 # Chai Wah Wu, Oct 26 2023

Formula

a(n) = Sum_{k=1..n} binomial(k+1,2) * floor(n/k).
G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1-x^k)^3 = 1/(1-x) * Sum_{k>=1} binomial(k+1,2) * x^k/(1-x^k).
a(n) = (A064602(n)+A024916(n))/2. - Chai Wah Wu, Oct 26 2023

A363628 Expansion of Sum_{k>0} (1/(1-x^k)^3 - 1).

Original entry on oeis.org

3, 9, 13, 24, 24, 47, 39, 69, 68, 96, 81, 153, 108, 165, 170, 222, 174, 292, 213, 342, 302, 363, 303, 523, 375, 492, 474, 615, 468, 766, 531, 783, 686, 810, 726, 1101, 744, 999, 938, 1248, 906, 1402, 993, 1413, 1306, 1437, 1179, 1901, 1314, 1773, 1562, 1938, 1488, 2238, 1698
Offset: 1

Views

Author

Seiichi Manyama, Jun 12 2023

Keywords

Crossrefs

Cf. A007503, A116963.
Cf. A007437, A069153, A363610.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, Binomial[# + 2, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 05 2023 *)
  • PARI
    a(n) = sumdiv(n, d, binomial(d+2, 2));

Formula

G.f.: Sum_{k>0} binomial(k+2,2) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d+2,2).

A363650 Expansion of Sum_{k>0} x^k/(1 - (k*x)^k)^3.

Original entry on oeis.org

1, 4, 7, 23, 16, 199, 29, 1445, 4420, 13271, 67, 751597, 92, 2585423, 66565486, 218693769, 154, 14527231822, 191, 399614708821, 4080186211018, 856004218103, 277, 2754664372347481, 1430511474609701, 908626846503767, 900580521111136750, 5626675967703843613, 436
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A363647, A363651, A363652.
Cf. A007437, A363642.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(n-n/#) * Binomial[# + 1, 2] &]; Array[a, 30] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(n-n/d)*binomial(d+1, 2));

Formula

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

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

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Dec 02 2016

Keywords

Comments

Inverse Moebius transform of hexagonal numbers (A000384).

Links

Crossrefs

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

Programs

  • Magma
    [0] cat [2*DivisorSigma(2, n) - DivisorSigma(1, n): n in [1..60]]; // Vincenzo Librandi, Dec 07 2016
  • Mathematica
    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 *)
  • PARI
    a(n) = if(n == 0, 0, my(f = factor(n)); 2 * sigma(f, 2) - sigma(f)); \\ Amiram Eldar, Dec 29 2024

Formula

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

A309732 Expansion of Sum_{k>=1} k^2 * x^k/(1 - x^k)^3.

Original entry on oeis.org

1, 7, 15, 38, 40, 108, 77, 188, 180, 290, 187, 600, 260, 560, 630, 888, 442, 1323, 551, 1620, 1218, 1364, 805, 3024, 1325, 1898, 1998, 3136, 1276, 4680, 1457, 4080, 2970, 3230, 3290, 7470, 2072, 4028, 4134, 8200, 2542, 9072, 2795, 7656, 7830, 5888, 3337, 14496, 4998, 9825, 7038
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 14 2019

Keywords

Comments

Dirichlet convolution of triangular numbers (A000217) with squares (A000290).
a(n) is n times half m, where m is the sum of all parts plus the total number of parts of the partitions of n into equal parts. - Omar E. Pol, Nov 30 2019

Links

Crossrefs

Cf. A000005, A000203, A000217, A000290, A007437, A034714, A034715, A038040, A064987, A152211, A309731, A244051.

Programs

  • Magma
    [n*(n*NumberOfDivisors(n) + DivisorSigma(1,n))/2:n in [1..51]]; // Marius A. Burtea, Nov 29 2019
  • Maple
    with(numtheory): seq(n*(n*tau(n)+sigma(n))/2, n=1..50); # Ridouane Oudra, Nov 28 2019
  • Mathematica
    nmax = 51; CoefficientList[Series[Sum[k^2 x^k/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
Table[DirichletConvolve[j (j + 1)/2, j^2, j, n], {n, 1, 51}]
Table[n (n DivisorSigma[0, n] + DivisorSigma[1, n])/2, {n, 1, 51}]
  • PARI
    a(n)=sumdiv(n, d, binomial(n/d+1,2)*d^2); \\ Andrew Howroyd, Aug 14 2019
  • PARI
    a(n)=n*(n*numdiv(n) + sigma(n))/2; \\ Andrew Howroyd, Aug 14 2019

Formula

G.f.: Sum_{k>=1} (k*(k + 1)/2) * x^k * (1 + x^k)/(1 - x^k)^3.
a(n) = n * (n * d(n) + sigma(n))/2.
Dirichlet g.f.: zeta(s-2) * (zeta(s-2) + zeta(s-1))/2.
a(n) = n*(A038040(n) + A000203(n))/2 = n*A152211(n)/2. - Omar E. Pol, Aug 17 2019
a(n) = Sum_{k=1..n} k*sigma(gcd(n,k)). - Ridouane Oudra, Nov 28 2019

A365007 a(n) = Sum_{d|n} (-1)^(n/d-1) * binomial(d+1,2).

Original entry on oeis.org

1, 2, 7, 6, 16, 17, 29, 22, 52, 42, 67, 57, 92, 79, 142, 86, 154, 143, 191, 146, 266, 189, 277, 217, 341, 262, 430, 279, 436, 402, 497, 342, 634, 444, 674, 507, 704, 553, 878, 562, 862, 766, 947, 677, 1222, 807, 1129, 857, 1254, 992, 1486, 942, 1432, 1250, 1622, 1079
Offset: 1

Views

Author

Seiichi Manyama, Oct 24 2023

Keywords

Links

Crossrefs

Partial sums give A366395.
Cf. A000593, A366813, A366814.
Cf. A007437.

Programs

  • Mathematica
    Table[DivisorSum[n, (-1)^(n/# - 1)*Binomial[# + 1, 2] &], {n, 56}] (* Michael De Vlieger, Oct 25 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (-1)^(n/d-1)*binomial(d+1, 2));

Formula

G.f.: -Sum_{k>=1} (-x)^k/(1-x^k)^3 = Sum_{k>=1} binomial(k+1,2) * x^k/(1+x^k).

A116913 Inverse Moebius transform of pentagonal numbers.

Original entry on oeis.org

1, 6, 13, 28, 36, 69, 71, 120, 130, 186, 177, 301, 248, 363, 378, 496, 426, 663, 533, 798, 734, 897, 783, 1245, 961, 1254, 1210, 1547, 1248, 1914, 1427, 2016, 1806, 2148, 1926, 2821, 2036, 2685, 2522, 3270, 2502, 3702, 2753, 3801, 3510, 3939, 3291, 5053, 3648
Offset: 1

Views

Author

Jonathan Vos Post, Mar 19 2006

Keywords

Links

Crossrefs

Cf. A000326 (pentagonal numbers), A000203, A002117.
Inverse Moebius transforms of polygonal numbers: A007437 (k=3), A001157 (k=4), this sequence (k=5), A278945 (k=6), A278947 (k=7).

Programs

  • Mathematica
    Table[Sum[d*(3d - 1)/2, {d, Divisors[n]}], {n, 101}] (* Indranil Ghosh, May 23 2017 *)
  • PARI
    a(n) = sumdiv(n, d, d*(3*d-1)/2); \\ Michel Marcus, Mar 25 2015
  • PARI
    a(n) = {my(f = factor(n)); (3 * sigma(f, 2) - sigma(f)) / 2;} \\ Amiram Eldar, Dec 29 2024

Formula

a(n) = Sum_{d|n} d*(3*d-1)/2.
G.f.: Sum_{k>=1} k*(3*k-1)/2*x^k/(1 - x^k). - Ilya Gutkovskiy, May 23 2017
From Amiram Eldar, Dec 29 2024: (Start)
a(n) = (3*sigma_2(n) - sigma(n)) / 2 = (3*A001157(n) - A000203(n)) / 2.
Dirichlet g.f.: zeta(s) * (3*zeta(s-2) - zeta(s-1))/2.
Sum_{k=1..n} a(k) ~ (zeta(3)/2) * n^3. (End)

Extensions

More terms from Michel Marcus, Mar 25 2015
Previous Showing 11-20 of 32 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.