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 20 results.

A339712 a(n) = Sum_{d|n} d^(d+n/d-1) * binomial(d+n/d-2, d-1).

Original entry on oeis.org

1, 5, 28, 273, 3126, 46948, 823544, 16781441, 387421948, 10000078446, 285311670612, 8916102176891, 302875106592254, 11112006865913416, 437893890382064056, 18446744074783625217, 827240261886336764178, 39346408075327954053967, 1978419655660313589123980
Offset: 1

Views

Author

Seiichi Manyama, Apr 24 2021

Keywords

Crossrefs

Cf. A157019, A157020, A324158, A324159, A338661, A339481, A339482, A343573.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(# + n/# - 1) * Binomial[# + n/# - 2, # - 1] &]; Array[a, 20] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = sumdiv(n, d, d^(d+n/d-1)*binomial(d+n/d-2, d-1));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=1, N, (k*x/(1-k*x^k))^k))

Formula

G.f.: Sum_{k >= 1} (k * x/(1 - k * x^k))^k.
If p is prime, a(p) = 1 + p^p.

A339482 a(n) = Sum_{d|n} d^(n-d+1) * binomial(d+n/d-2, d-1).

Original entry on oeis.org

1, 3, 4, 21, 6, 346, 8, 4617, 13132, 80696, 12, 4903847, 14, 40410966, 756336736, 2416181265, 18, 306560794753, 20, 6941876836216, 132964265599502, 34522735212626, 24, 116720277621236637, 33378601074218776, 51185893450298400, 60788365423272068968
Offset: 1

Views

Author

Seiichi Manyama, Apr 24 2021

Keywords

Crossrefs

Cf. A157019, A157020, A324158, A324159, A338661, A339481, A339712, A343573.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(n - # + 1) * Binomial[# + n/# - 2, # - 1] &]; Array[a, 30] (* Amiram Eldar, Apr 25 2021 *)
  • PARI
    a(n) = sumdiv(n, d, d^(n-d+1)*binomial(d+n/d-2, d-1));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, k*(x/(1-(k*x)^k))^k))

Formula

G.f.: Sum_{k >= 1} k * (x/(1 - (k * x)^k))^k.
If p is prime, a(p) = 1 + p.

A309634 G.f.: x * Sum_{k>=1} x^k / (1 - x^k)^a(k).

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 3, 1, 3, 2, 4, 1, 5, 1, 5, 5, 4, 1, 7, 1, 9, 8, 6, 1, 7, 6, 7, 8, 14, 1, 18, 1, 9, 12, 6, 23, 17, 1, 9, 17, 17, 1, 35, 1, 31, 41, 8, 1, 23, 29, 24, 12, 44, 1, 33, 47, 49, 30, 16, 1, 61, 1, 20, 120, 40, 84, 105, 1, 35, 23, 85, 1, 68, 1, 19, 115, 88, 151, 160, 1
Offset: 1

Views

Author

Ilya Gutkovskiy, Aug 10 2019

Keywords

Links

Crossrefs

Cf. A028815 (positions of 1's), A157019, A309633.

Programs

  • Maple
    a:= proc(n) option remember; uses numtheory;
      add(binomial((n-1)/d+a(d)-2, a(d)-1), d=divisors(n-1))
    end:
seq(a(n), n=1..80);  # Alois P. Heinz, Jan 27 2025
  • Mathematica
    a[n_] := a[n] = SeriesCoefficient[x Sum[x^k/(1 - x^k)^a[k], {k, 1, n - 1}], {x, 0, n}]; Table[a[n], {n, 1, 80}]
a[n_] := a[n] = Sum[Binomial[(n - 1)/d + a[d] - 2, a[d] - 1], {d, Divisors[n - 1]}]; a[1] = 0; Table[a[n], {n, 1, 80}]
  • PARI
    seq(n)={my(v=vector(n)); v[2]=1; for(n=2, #v-1, v[n+1] = sumdiv(n, d, binomial(n/d + v[d] - 2, v[d] - 1))); v} \\ Andrew Howroyd, Aug 10 2019

Formula

a(1) = 0; a(n+1) = Sum_{d|n} binomial(n/d+a(d)-2,a(d)-1).

A157028 Triangle read by rows, A007318 * A156348.

Original entry on oeis.org

1, 2, 1, 4, 2, 1, 8, 5, 3, 1, 16, 12, 6, 4, 1, 32, 28, 13, 10, 5, 1, 64, 64, 33, 20, 15, 6, 1, 128, 144, 84, 39, 35, 21, 7, 1, 256, 320, 202, 88, 70, 56, 28, 8, 1, 512, 704, 468, 228, 131, 126, 84, 36, 9, 1, 1024, 1536, 1071, 600, 260, 252, 210, 120, 45, 10, 1
Offset: 1

Views

Author

Gary W. Adamson & Mats Granvik, Feb 21 2009

Keywords

Comments

Row sums = A157029: (1, 3, 7, 17, 39, 89, 203, 459,...).

Examples

			1;
2, 1;
4, 2, 1;
8, 5, 3, 1;
16, 12, 6, 4, 1;
32, 28, 13, 10, 5, 1;
64, 64, 33, 20, 15, 6, 1;
128, 144, 84, 39, 35, 21, 7, 1;
256, 320, 202, 88, 70, 56, 28, 8, 1;
512, 704, 468, 228, 131, 126, 84, 36, 9, 1;
1024, 1536, 1071, 600, 260, 252, 210, 120, 45, 10, 1;
2048, 3328, 2441, 1495, 605, 468, 462, 330, 165, 55, 11, 1;
4096, 7168, 5532, 3508, 1595, 864, 924, 792, 495, 220, 66, 12, 1;
...

Crossrefs

Cf. A156348, A157019, A157029

Formula

Triangle read by rows, binomial transform of A156348.

A157030 Triangle read by rows, A156834 * A054521.

Original entry on oeis.org

1, 2, 0, 2, 1, 0, 4, 0, 1, 0, 2, 1, 1, 1, 0, 8, 3, 0, 0, 1, 0, 2, 1, 1, 1, 1, 1, 0, 10, 0, 5, 0, 1, 0, 1, 0, 8, 7, 0, 1, 1, 0, 1, 1, 0, 12, 5, 6, 5, 0, 0, 10, 1, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 34, 10, 10, 0, 7, 0, 10, 0, 0, 1, 0
Offset: 1

Views

Author

Gary W. Adamson and Mats Granvik, Feb 21 2009

Keywords

Comments

Left border = A157019: (1, 2, 2, 4, 2, 8, 2, 10,...).

Examples

			First few rows of the triangle =
1;
2, 0;
2, 1, 0;
4, 0, 1, 0;
2, 1, 1, 1, 0;
8, 3, 0, 0, 1, 0;
2, 1, 1, 1, 1, 1, 0;
10, 0, 5, 0, 1, 0, 1, 0;
8, 7, 0, 1, 1, 0, 1, 1, 0;
12, 5, 6, 5, 0, 0, 10, 1, 0;
2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0;
34, 10, 10, 0, 7, 0, 1, 0, 0, 0, 1, 0;
2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0;
...

Crossrefs

Cf. A156834 (row sums), A054521.

Formula

Triangle read by rows, A156834 * A054521; as infinite lower triangular matrices.

A358410 a(n) = Sum_{d|n} (d + n/d - 2)!/(d - 1)!.

Original entry on oeis.org

1, 2, 3, 9, 25, 130, 721, 5069, 40333, 363006, 3628801, 39917607, 479001601, 6227025848, 87178291591, 1307674408449, 20922789888001, 355687428461452, 6402373705728001, 121645100412461861, 2432902008176660217, 51090942171749356812, 1124000727777607680001
Offset: 1

Views

Author

Seiichi Manyama, Nov 14 2022

Keywords

Crossrefs

Cf. A157019, A358280, A358389.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (# + n/# - 2)!/(# - 1)! &]; Array[a, 23] (* Amiram Eldar, Aug 30 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (d+n/d-2)!/(d-1)!);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=1, N, (k-1)!*(x/(1-x^k))^k))

Formula

G.f.: Sum_{k>0} (k-1)! * (x/(1 - x^k))^k.
If p is prime, a(p) = 1 + (p-1)!.

A330020 Expansion of e.g.f. Sum_{k>=1} x^k / (k! * (1 - x^k)^k).

Original entry on oeis.org

1, 3, 7, 49, 121, 2161, 5041, 127681, 725761, 12852001, 39916801, 2917918081, 6227020801, 392423391361, 4740319584001, 122053759027201, 355687428096001, 57808258040332801, 121645100408832001, 18854997267794688001, 289799177540640768001, 7306005040298918553601
Offset: 1

Views

Author

Ilya Gutkovskiy, Nov 27 2019

Keywords

Crossrefs

Cf. A057625, A157019, A327578, A327579.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[Sum[x^k/(k! (1 - x^k)^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]! // Rest
a[n_] := n! Sum[(d + n/d - 2)!/(d! (d - 1)! (n/d - 1)!), {d, Divisors[n]}]; Table[a[n], {n, 1, 22}]

Formula

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

A360801 Expansion of Sum_{k>0} (x / (1 - 2 * x^k))^k.

Original entry on oeis.org

1, 3, 5, 13, 17, 51, 65, 169, 281, 603, 1025, 2373, 4097, 8655, 16685, 33969, 65537, 134151, 262145, 530269, 1050481, 2108439, 4194305, 8420201, 16778337, 33607707, 67120565, 134338493, 268435457, 537151131, 1073741825, 2148024289, 4295035145, 8591048739
Offset: 1

Views

Author

Seiichi Manyama, Feb 21 2023

Keywords

Crossrefs

Cf. A157019, A217670, A324158.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 2^(n/# - 1) * Binomial[# + n/# - 2, # - 1] &]; Array[a, 30] (* Amiram Eldar, Aug 02 2023 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (x/(1-2*x^k))^k))
  • PARI
    a(n) = sumdiv(n, d, 2^(n/d-1)*binomial(d+n/d-2, d-1));

Formula

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

A360802 Expansion of Sum_{k>0} (x / (1 - (2 * x)^k))^k.

Original entry on oeis.org

1, 3, 5, 17, 17, 105, 65, 449, 641, 1953, 1025, 16257, 4097, 37761, 93185, 247809, 65537, 1499649, 262145, 6596609, 8847361, 13654017, 4194305, 210026497, 90177537, 251764737, 833880065, 2659418113, 268435457, 18345328641, 1073741825, 53553922049, 75438751745
Offset: 1

Views

Author

Seiichi Manyama, Feb 21 2023

Keywords

Crossrefs

Cf. A157019, A339481.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, 2^(n-#) * Binomial[# + n/# - 2, # - 1] &]; Array[a, 30] (* Amiram Eldar, Aug 02 2023 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, (x/(1-(2*x)^k))^k))
  • PARI
    a(n) = sumdiv(n, d, 2^(n-d)*binomial(d+n/d-2, d-1));

Formula

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

A382512 Expansion of Sum_{p prime} x^p / (1 - x^p)^p.

Original entry on oeis.org

0, 1, 1, 2, 1, 6, 1, 4, 6, 10, 1, 16, 1, 14, 30, 8, 1, 30, 1, 45, 56, 22, 1, 48, 70, 26, 45, 98, 1, 196, 1, 16, 132, 34, 420, 96, 1, 38, 182, 350, 1, 588, 1, 308, 615, 46, 1, 160, 924, 740, 306, 481, 1, 198, 2002, 1744, 380, 58, 1, 1605, 1, 62, 3234, 32, 3640
Offset: 1

Views

Author

Ilya Gutkovskiy, Mar 30 2025

Keywords

Crossrefs

Cf. A001221, A069359, A157019, A322078, A373458, A373459.

Programs

  • Mathematica
    nmax = 65; CoefficientList[Series[Sum[x^Prime[k]/(1 - x^Prime[k])^Prime[k], {k, 1, nmax}], {x, 0, nmax}], x] // Rest

Formula

a(n) = Sum_{p|n, p prime} binomial(n/p+p-2, p-1).
Previous Showing 11-20 of 20 results.

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

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