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 1-6 of 6 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)

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)

A366971 a(n) = Sum_{k=3..n} binomial(k,3) * floor(n/k).

Original entry on oeis.org

0, 0, 1, 5, 15, 36, 71, 131, 216, 346, 511, 756, 1042, 1441, 1907, 2527, 3207, 4128, 5097, 6371, 7737, 9442, 11213, 13538, 15848, 18734, 21744, 25423, 29077, 33743, 38238, 43818, 49440, 56104, 62694, 70979, 78749, 88154, 97580, 108790, 119450, 132680, 145021, 159974
Offset: 1

Views

Author

Seiichi Manyama, Oct 30 2023

Keywords

Crossrefs

Partial sums of A363607.
Cf. A366968, A366969, A366970.
Cf. A024916, A064602, A064603, A366967.

Programs

  • PARI
    a(n) = sum(k=3, n, binomial(k, 3)*(n\k));
  • Python
    from math import isqrt, comb
def A366971(n): return -comb((s:=isqrt(n))+1,4)*(s+1)+sum(comb((q:=n//w)+1,4)+(q+1)*comb(w,3) for w in range(1,s+1)) # Chai Wah Wu, Oct 30 2023

Formula

G.f.: 1/(1-x) * Sum_{k>=1} x^(3*k)/(1-x^k)^4 = 1/(1-x) * Sum_{k>=3} binomial(k,3) * x^k/(1-x^k).
a(n) = (A064603(n) - 3*A064602(n) + 2*A024916(n))/6. - Chai Wah Wu, Oct 30 2023

A363608 Expansion of Sum_{k>0} x^(4*k)/(1-x^k)^5.

Original entry on oeis.org

0, 0, 0, 1, 5, 15, 35, 71, 126, 215, 330, 511, 715, 1036, 1370, 1891, 2380, 3201, 3876, 5061, 6020, 7645, 8855, 11207, 12655, 15665, 17676, 21512, 23751, 29000, 31465, 37851, 41250, 48756, 52400, 62602, 66045, 77691, 82966, 96521, 101270, 118966, 123410, 143397
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2023

Keywords

Links

Crossrefs

Cf. A013663, A069153, A363607.
Cf. A000203, A001157, A001158, A001159.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, Binomial[#, 4] &]; 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)^5)))
  • PARI
    a(n) = my(f = factor(n)); (sigma(f, 4) - 6*sigma(f, 3) + 11*sigma(f, 2) - 6*sigma(f)) / 24; \\ Amiram Eldar, Dec 30 2024

Formula

G.f.: Sum_{k>0} binomial(k,4) * x^k/(1 - x^k).
a(n) = Sum_{d|n} binomial(d,4).
From Amiram Eldar, Dec 30 2024: (Start)
a(n) = (sigma_4(n) - 6*sigma_3(n) + 11*sigma_2(n) - 6*sigma_1(n)) / 24.
Dirichlet g.f.: zeta(s) * (zeta(s-4) - 6*zeta(s-3) + 11*zeta(s-2) - 6*zeta(s-1)) / 24.
Sum_{k=1..n} a(k) ~ (zeta(5)/120) * n^5. (End)

A363617 Expansion of Sum_{k>0} x^(3*k)/(1+x^k)^4.

Original entry on oeis.org

0, 0, 1, -4, 10, -19, 35, -60, 85, -110, 165, -243, 286, -329, 466, -620, 680, -751, 969, -1254, 1366, -1375, 1771, -2323, 2310, -2314, 3010, -3609, 3654, -3734, 4495, -5580, 5622, -5304, 6590, -8115, 7770, -7467, 9426, -11190, 10660, -10498, 12341, -14623, 14740, -13409, 16215, -20179, 18459, -17410, 21506
Offset: 1

Views

Author

Seiichi Manyama, Jun 11 2023

Keywords

Links

Crossrefs

Cf. A363022, A363618.
Cf. A363607.

Programs

  • Mathematica
    a[n_] := -DivisorSum[n, (-1)^#*Binomial[#, 3] &]; 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)^4)))
  • PARI
    a(n) = -sumdiv(n, d, (-1)^d*binomial(d, 3));

Formula

G.f.: -Sum_{k>0} binomial(k,3) * (-x)^k/(1 - x^k).
a(n) = -Sum_{d|n} (-1)^d * binomial(d,3).
Showing 1-6 of 6 results.

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