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.

A116963 Inverse Moebius transform of the shifted tetrahedral numbers.

Original entry on oeis.org

4, 14, 24, 49, 60, 118, 124, 214, 244, 356, 368, 608, 564, 814, 896, 1183, 1144, 1668, 1544, 2162, 2168, 2678, 2604, 3698, 3336, 4228, 4304, 5344, 4964, 6732, 5988, 7728, 7528, 8924, 8616, 11297, 9884, 12214, 12064, 14668, 13248, 17132, 15184, 18928, 18412, 21038
Offset: 1

Views

Author

Jonathan Vos Post, Mar 31 2006

Keywords

Examples

			a(12) = ((1+1)*(1+2)*(1+3)/6) + ((2+1)*(2+2)*(2+3)/6) + ((3+1)*(3+2)*(3+3)/6) + ((4+1)*(4+2)*(4+3)/6) + ((6+1)*(6+2)*(6+3)/6) + ((12+1)*(12+2)*(12+3)/6) = 4 + 10 + 20 + 35 + 84 + 455 = 608.
a(13) = ((1+1)*(1+2)*(1+3)/6) + ((13+1)*(13+2)*(13+3)/6) = 4 + 560 = 564.

Links

Crossrefs

See also: A007437 (inverse Moebius transform of triangular numbers).
Cf. A000292, A007437, A007503.
Cf. A013662, A059358, A363604, A363607, A363611.
Cf. A000005, A000203, A001157, A001158.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, Binomial[# + 3, 3] &]; Array[a, 50] (* Amiram Eldar, Jul 05 2023 *)
  • PARI
    my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, 1/(1-x^k)^4-1)) \\ Seiichi Manyama, Jun 12 2023

Formula

a(n) = Sum_{d|n} (d+1)*(d+2)*(d+3)/6 = Sum_{d|n} A000292(d+1).
G.f.: Sum_{k>0} (1/(1-x^k)^4 - 1). - Seiichi Manyama, Jun 12 2023
From Amiram Eldar, Dec 30 2024: (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)

A363604 Expansion of Sum_{k>0} x^(2*k)/(1-x^k)^4.

Original entry on oeis.org

0, 1, 4, 11, 20, 40, 56, 95, 124, 186, 220, 336, 364, 512, 584, 775, 816, 1129, 1140, 1526, 1600, 1992, 2024, 2720, 2620, 3290, 3400, 4176, 4060, 5280, 4960, 6231, 6208, 7362, 7216, 9195, 8436, 10280, 10248, 12270, 11480, 14432, 13244, 16192, 15884, 18240
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2023

Keywords

Links

Crossrefs

Cf. A000292, A032741, A065608, A069153, A363605, A363606.
Cf. A059358, A363607, A363611.
Cf. A000203, A001158, A013662, A092348.

Programs

  • Mathematica
    a[n_] := (DivisorSigma[3, n] - DivisorSigma[1, n])/6; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)
  • PARI
    my(N=50, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^(2*k)/(1-x^k)^4)))
  • PARI
    a(n) = my(f = factor(n)); (sigma(f, 3) - sigma(f))/6; \\ Amiram Eldar, Dec 30 2024

Formula

a(n) = (sigma_3(n) - sigma(n))/6 = A092348(n)/6.
G.f.: Sum_{k>0} binomial(k+1,3) * x^k/(1 - x^k).
From Amiram Eldar, Dec 30 2024: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s-3) - zeta(s-1)) / 6.
Sum_{k=1..n} a(k) ~ (zeta(4)/24) * n^4. (End)

A363607 Expansion of Sum_{k>0} x^(3*k)/(1-x^k)^4.

Original entry on oeis.org

0, 0, 1, 4, 10, 21, 35, 60, 85, 130, 165, 245, 286, 399, 466, 620, 680, 921, 969, 1274, 1366, 1705, 1771, 2325, 2310, 2886, 3010, 3679, 3654, 4666, 4495, 5580, 5622, 6664, 6590, 8285, 7770, 9405, 9426, 11210, 10660, 13230, 12341, 14953, 14740, 16951, 16215, 20181
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2023

Keywords

Links

Crossrefs

Cf. A013662, A069153, A363608.
Cf. A059358, A363604, A363611.
Cf. A000203, A001157, A001158.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, Binomial[#, 3] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)
  • PARI
    my(N=50, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/(1-x^k)^4)))
  • PARI
    a(n) = my(f = factor(n)); (sigma(f, 3) - 3*sigma(f, 2) + 2*sigma(f)) / 6; \\ Amiram Eldar, Dec 30 2024

Formula

G.f.: Sum_{k>0} binomial(k,3) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d,3).
From Amiram Eldar, Dec 30 2024: (Start)
a(n) = (sigma_3(n) - 3*sigma_2(n) + 2*sigma_1(n)) / 6.
Dirichlet g.f.: zeta(s) * (zeta(s-3) - 3*zeta(s-2) + 2*zeta(s-1)) / 6.
Sum_{k=1..n} a(k) ~ (zeta(4)/24) * n^4. (End)

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)

A363616 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, 80, -120, 176, -220, 266, -368, 491, -560, 634, -816, 1050, -1160, 1210, -1540, 1982, -2028, 2080, -2656, 3192, -3276, 3380, -4060, 4986, -5080, 4896, -6008, 7345, -7140, 6954, -8656, 10224, -9880, 9796, -11480, 13552, -13668, 12650
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2023

Keywords

Links

Crossrefs

Cf. A002129, A128315, A138503, A321543, A325940, A363611, A363615.

Programs

  • Magma
    A363616:= func< n | (&+[(-1)^d*Binomial(d-1,3): d in Divisors(n)]) >;
[A363616(n): n in [1..60]]; // G. C. Greubel, Jun 22 2024
  • Mathematica
    a[n_] := DivisorSum[n, (-1)^# * Binomial[# - 1, 3] &]; Array[a, 50] (* Amiram Eldar, Jul 25 2023 *)
  • PARI
    my(N=50, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/(1+x^k)^4)))
  • PARI
    a(n) = sumdiv(n, d, (-1)^d*binomial(d-1, 3));
  • SageMath
    def A363616(n): return sum(0^(n%j)*(-1)^j*binomial(j-1,3) for j in range(4, n+1))
[A363616(n) for n in range(1,61)] # G. C. Greubel, Jun 22 2024

Formula

G.f.: Sum_{k>0} binomial(k-1,3) * (-x)^k/(1 - x^k).
a(n) = Sum_{d|n} (-1)^d * binomial(d-1,3).
a(n) = A128315(n, 4), for n >= 4. - G. C. Greubel, Jun 22 2024
a(n) = -(A138503(n) - 6*A321543(n) + 11*A002129(n) - 6*A048272(n)) / 6. - Amiram Eldar, Jan 04 2025
Showing 1-5 of 5 results.

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