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.

Showing 1-5 of 5 results.

A363611 Expansion of Sum_{k>0} x^(4*k)/(1-x^k)^4.

Original entry on oeis.org

0, 0, 0, 1, 4, 10, 20, 36, 56, 88, 120, 176, 220, 306, 368, 491, 560, 746, 816, 1058, 1160, 1450, 1540, 1982, 2028, 2520, 2656, 3232, 3276, 4116, 4060, 4986, 5080, 6016, 6008, 7457, 7140, 8586, 8656, 10232, 9880, 12116, 11480, 13792, 13668, 15730, 15180, 18652, 17316, 20536
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2023

Keywords

Links

Crossrefs

Cf. A000005, A000203, A001157, A001158, A013662, A065608, A363610.
Cf. A059358, A363604, A363607.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, Binomial[# - 1, 3] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)
  • PARI
    my(N=60, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/(1-x^k)^4)))
  • PARI
    a(n) = my(f = factor(n)); (sigma(f, 3) - 6*sigma(f, 2) + 11*sigma(f) - 6*numdiv(f)) / 6; \\ Amiram Eldar, Jan 01 2025

Formula

G.f.: Sum_{k>0} binomial(k-1,3) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d-1,3).
From Amiram Eldar, Jan 01 2025: (Start)
a(n) = (sigma_3(n) - 6*sigma_2(n) + 11*sigma_1(n) - 6*sigma_0(n)) / 6.
Dirichlet g.f.: zeta(s) * (zeta(s-3) - 6*zeta(s-2) + 11*zeta(s-1) - 6*zeta(s)) / 6.
Sum_{k=1..n} a(k) ~ (zeta(4)/24) * n^4. (End)

A363615 Expansion of Sum_{k>0} x^(3*k)/(1+x^k)^3.

Original entry on oeis.org

0, 0, 1, -3, 6, -9, 15, -24, 29, -30, 45, -67, 66, -63, 98, -129, 120, -117, 153, -204, 206, -165, 231, -341, 282, -234, 354, -417, 378, -354, 435, -594, 542, -408, 582, -770, 630, -513, 770, -966, 780, -702, 861, -1071, 1072, -759, 1035, -1527, 1143, -930, 1346
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2023

Keywords

Links

Crossrefs

Cf. A002129, A048272, A128315, A325940, A321543, A363610, A363616.

Programs

  • Magma
    A363615:= func< n | -(&+[(-1)^d*Binomial(d-1,2): d in Divisors(n)]) >;
[A363615(n): n in [1..60]]; // G. C. Greubel, Jun 22 2024
  • Mathematica
    a[n_] := -DivisorSum[n, (-1)^#*Binomial[# - 1, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 18 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) = -sumdiv(n, d, (-1)^d*binomial(d-1, 2));
  • SageMath
    def A363615(n): return sum(0^(n%j)*(-1)^(j+1)*binomial(j-1,2) for j in range(1, n+1))
[A363615(n) for n in range(1,61)] # G. C. Greubel, Jun 22 2024

Formula

G.f.: -Sum_{k>0} binomial(k-1,2) * (-x)^k/(1 - x^k).
a(n) = -Sum_{d|n} (-1)^d * binomial(d-1,2).
a(n) = A128315(n, 3), for n >= 3. - G. C. Greubel, Jun 22 2024
a(n) = (A321543(n) - 3*A002129(n) + 2*A048272(n)) / 2. - Amiram Eldar, Jan 04 2025

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).

A363644 Expansion of Sum_{k>0} x^(3*k)/(1 - k*x^k)^3.

Original entry on oeis.org

0, 0, 1, 3, 6, 11, 15, 27, 29, 60, 45, 145, 66, 318, 146, 789, 120, 2199, 153, 4890, 1406, 11730, 231, 34175, 426, 67884, 20738, 163956, 378, 453341, 435, 882153, 295742, 1966608, 10230, 5656484, 630, 10027674, 3897938, 23069916, 780, 63648517, 861, 113050536
Offset: 1

Views

Author

Seiichi Manyama, Jun 13 2023

Keywords

Crossrefs

Cf. A363639, A363642, A363643.
Cf. A363610, A363652.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (n/#)^(# - 3)*Binomial[# - 1, 2] &]; Array[a, 50] (* Amiram Eldar, Jul 18 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (n/d)^(d-3)*binomial(d-1, 2));

Formula

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

A366970 a(n) = Sum_{k=3..n} binomial(k-1,2) * floor(n/k).

Original entry on oeis.org

0, 0, 1, 4, 10, 21, 36, 60, 89, 131, 176, 245, 311, 404, 502, 631, 751, 926, 1079, 1295, 1501, 1756, 1987, 2330, 2612, 2978, 3332, 3779, 4157, 4707, 5142, 5736, 6278, 6926, 7508, 8336, 8966, 9785, 10555, 11533, 12313, 13427, 14288, 15449, 16521, 17742, 18777, 20306
Offset: 1

Views

Author

Seiichi Manyama, Oct 30 2023

Keywords

Crossrefs

Partial sums of A363610.
Cf. A006218, A024916, A064602, A366968, A366969, A366971.

Programs

  • PARI
    a(n) = sum(k=3, n, binomial(k-1, 2)*(n\k));
  • Python
    from math import isqrt
def A366970(n): return (-(s:=isqrt(n))*(s*(s**2-(s<<1)-1)+8)+sum(((q:=n//w)+1)*(q*(q-4)+3*(w**2-3*w+4)) for w in range(1,s+1)))//6 # Chai Wah Wu, Oct 30 2023

Formula

G.f.: 1/(1-x) * Sum_{k>=1} x^(3*k)/(1-x^k)^3 = 1/(1-x) * Sum_{k>=3} binomial(k-1,2) * x^k/(1-x^k).
a(n) = (A064602(n)-3*A024916(n))/2 + A006218(n). - Chai Wah Wu, Oct 30 2023
Showing 1-5 of 5 results.

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