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 1-10 of 20 results. Next

A073570 G.f.: Sum_{n >= 1} x^n/(1-x^n)^5.

Original entry on oeis.org

1, 6, 16, 41, 71, 147, 211, 371, 511, 791, 1002, 1547, 1821, 2596, 3146, 4247, 4846, 6627, 7316, 9681, 10852, 13657, 14951, 19427, 20546, 25577, 27916, 34096, 35961, 44912, 46377, 56607, 59922, 70896, 74096, 90278, 91391, 108591, 113766, 133421
Offset: 1

Views

Author

Vladeta Jovovic, Aug 31 2002

Keywords

Comments

Inverse Moebius transform of pentatope numbers. - Jonathan Vos Post, Mar 31 2006

Links

Crossrefs

Cf. A000005, A000332, A007437, A013663, A059358, A101289.
Cf. A000203, A001157, A001158, A001159.

Programs

  • Mathematica
    Table[(DivisorSigma[4,n]+6*DivisorSigma[3,n]+11*DivisorSigma[2,n]+ 6*DivisorSigma[ 1,n])/24,{n,40}] (* Harvey P. Dale, Aug 08 2013 *)
  • PARI
    a(n) = sumdiv(n, d, binomial(d+3, 4)); \\ Seiichi Manyama, Apr 19 2021
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+3, 4)*x^k/(1-x^k))) \\ Seiichi Manyama, Apr 19 2021
  • 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

a(n) = (1/24) * (sigma_4(n) + 6*sigma_3(n) + 11*sigma_2(n) + 6*sigma_1(n)).
a(n) = Sum_{d|n} (d+1)*(d+2)*(d+3)*(d+4)/24 = Sum_{d|n} C(d+3,4) = Sum_{d|n} A000332(d+3). - Jonathan Vos Post, Mar 31 2006. Corrected by Joshua Zucker, May 04 2007
From Amiram Eldar, Dec 30 2024: (Start)
Dirichlet g.f.: zeta(s) * (zeta(s-4) + 6*zeta(s-3) + 11*zeta(s-2) + 6*zeta(s-2)) / 24.
Sum_{k=1..n} a(k) ~ (zeta(5)/120) * n^5. (End)

Extensions

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 31 2007

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)

A332508 a(n) = Sum_{d|n} binomial(n+d-2, n-1).

Original entry on oeis.org

1, 3, 7, 25, 71, 280, 925, 3561, 12916, 49346, 184757, 710255, 2704157, 10427747, 40119781, 155288897, 601080391, 2334714319, 9075135301, 35352181116, 137846759282, 538302226628, 2104098963721, 8233718962365, 32247603703576, 126412458920775, 495918551104687
Offset: 1

Views

Author

Ilya Gutkovskiy, Feb 14 2020

Keywords

Links

Crossrefs

Cf. A000005, A000203, A007437, A059358, A073570, A101289, A308814, A332470.

Programs

  • Mathematica
    Table[DivisorSum[n, Binomial[n + # - 2, n - 1] &], {n, 1, 27}]
Table[SeriesCoefficient[Sum[x^k/(1 - x^k)^n, {k, 1, n}], {x, 0, n}], {n, 1, 27}]
  • PARI
    a(n) = sumdiv(n, d, binomial(n+d-2, n-1)); \\ Michel Marcus, Feb 14 2020

Formula

a(n) = [x^n] Sum_{k>=1} x^k / (1 - x^k)^n.
a(n) ~ 4^(n-1) / sqrt(Pi*n). - Vaclav Kotesovec, Aug 04 2022

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)

A117108 Moebius transform of tetrahedral numbers.

Original entry on oeis.org

1, 3, 9, 16, 34, 43, 83, 100, 155, 182, 285, 292, 454, 473, 636, 696, 968, 929, 1329, 1304, 1678, 1735, 2299, 2136, 2890, 2818, 3489, 3484, 4494, 4052, 5455, 5168, 6250, 6168, 7652, 6988, 9138, 8547, 10196, 9840, 12340, 10954, 14189, 13140, 15380, 14993, 18423
Offset: 1

Views

Author

Steve Butler, Apr 18 2006

Keywords

Comments

Partial sums of a(n) give A015634(n).
See also A059358, A116963 (applied to shifted version of tetrahedral numbers), inverse Moebius transform of tetrahedral numbers. - Jonathan Vos Post, Apr 20 2006

Examples

			a(2) = 3 because of the triples (1,1,1), (1,1,2), (1,2,2).

Links

Crossrefs

Cf. A000292 (tetrahedral numbers), A007438, A008683, A015634 (partial sums), A059358, A116963, A117109, A343544.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, MoebiusMu[n/#]*Binomial[# + 2, 3] &]; Array[a, 50] (* Amiram Eldar, Jun 07 2025 *)
  • PARI
    a(n) = sumdiv(n, d, binomial(d+2, 3)*moebius(n/d)); \\ Michel Marcus, Nov 04 2018

Formula

a(n) = |{(x,y,z) : 1 <= x <= y <= z <= n, gcd(x,y,z,n) = 1}|.
G.f.: Sum_{k>=1} mu(k) * x^k / (1 - x^k)^4. - Ilya Gutkovskiy, Feb 13 2020

Extensions

Offset changed to 1 by Ilya Gutkovskiy, Feb 13 2020

A343544 a(n) = n * Sum_{d|n} binomial(d+2,3)/d.

Original entry on oeis.org

1, 6, 13, 32, 40, 94, 91, 184, 204, 320, 297, 612, 468, 770, 850, 1184, 986, 1752, 1349, 2280, 2114, 2662, 2323, 4184, 3125, 4264, 4266, 5740, 4524, 7660, 5487, 8352, 7546, 9180, 8470, 13212, 9176, 12654, 12194, 16640, 12382, 19628, 14233, 20724, 19590, 22034, 18471, 30416, 21462
Offset: 1

Views

Author

Seiichi Manyama, Apr 19 2021

Keywords

Links

Crossrefs

Cf. A000203, A038040, A059358, A309731, A343545, A343546, A343547.

Programs

  • Maple
    f:= n -> n/6*add((d+1)*(d+2),d=numtheory:-divisors(n)):
map(f, [$1..100]); # Robert Israel, Apr 26 2021
  • Mathematica
    a[n_] := n * DivisorSum[n, Binomial[# + 2, 3]/# &]; Array[a, 50] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = n*sumdiv(n, d, binomial(d+2, 3)/d);
  • PARI
    my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, binomial(k+2, 3)*x^k/(1-x^k)^2))

Formula

G.f.: Sum_{k>=1} k * x^k/(1 - x^k)^4 = Sum_{k>=1} binomial(k+2,3) * x^k/(1 - x^k)^2.

A343548 a(n) = Sum_{d|n} binomial(d+n-1,n).

Original entry on oeis.org

1, 4, 11, 41, 127, 498, 1717, 6610, 24366, 93391, 352717, 1358826, 5200301, 20097076, 77562773, 300786339, 1166803111, 4539163784, 17672631901, 68933291834, 269129233484, 1052113994124, 4116715363801, 16124221819056, 63205303242628, 247961973949228, 973469736360283
Offset: 1

Views

Author

Seiichi Manyama, Apr 19 2021

Keywords

Links

Crossrefs

Cf. A000005, A000203, A007437, A059358, A073570, A101289, A332508, A343549.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, Binomial[# + n - 1, n] &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = sumdiv(n, d, binomial(d+n-1, n));

Formula

a(n) = [x^n] Sum_{k>=1} x^k/(1 - x^k)^(n+1).
a(n) = [x^n] Sum_{k>=1} binomial(k+n-1,n) * x^k/(1 - x^k).

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)

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)

A365409 a(n) = Sum_{k=1..n} binomial(floor(n/k)+3,4).

Original entry on oeis.org

1, 6, 17, 42, 78, 149, 234, 379, 555, 815, 1102, 1557, 2013, 2662, 3388, 4349, 5319, 6695, 8026, 9846, 11712, 14027, 16328, 19503, 22464, 26200, 30030, 34759, 39255, 45221, 50678, 57623, 64465, 72579, 80469, 90665, 99805, 111020, 122146, 135566, 147908, 163638
Offset: 1

Views

Author

Seiichi Manyama, Oct 23 2023

Keywords

Crossrefs

Partial sums of A059358.
Cf. A006218, A024916, A064602, A064603, A364970, A365439.

Programs

  • PARI
    a(n) = sum(k=1, n, binomial(n\k+3, 4));
  • Python
    from math import isqrt, comb
def A365409(n): return -(s:=isqrt(n))**2*comb(s+3,3)+sum((q:=n//k)*((comb(k+2,3)<<2)+comb(q+3,3)) for k in range(1,s+1))>>2 # Chai Wah Wu, Oct 26 2023

Formula

a(n) = Sum_{k=1..n} binomial(k+2,3) * floor(n/k).
G.f.: 1/(1-x) * Sum_{k>=1} x^k/(1-x^k)^4 = 1/(1-x) * Sum_{k>=1} binomial(k+2,3) * x^k/(1-x^k).
a(n) = (A064603(n)+3*A064602(n)+2*A024916(n))/6. - Chai Wah Wu, Oct 26 2023
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