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.

A360783 Expansion of Sum_{k>=0} x^k / (1 - k*x^3)^(k+1).

Original entry on oeis.org

1, 1, 1, 1, 3, 7, 13, 24, 55, 133, 301, 678, 1639, 4120, 10253, 25591, 65869, 173551, 459493, 1225379, 3325123, 9162046, 25451181, 71296499, 202144225, 579612934, 1675822453, 4885178596, 14376297345, 42690792651, 127757371105, 385241085261, 1170960103855
Offset: 0

Views

Author

Seiichi Manyama, Feb 20 2023

Keywords

Crossrefs

Cf. A000248, A360782.
Cf. A000930.

Programs

  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-k*x^3)^(k+1)))
  • PARI
    a(n) = sum(k=0, n\3, (n-3*k)^k*binomial(n-2*k, k));

Formula

a(n) = Sum_{k=0..floor(n/3)} (n-3*k)^k * binomial(n-2*k,k).

A360787 Expansion of Sum_{k>=0} x^k / (1 - (k*x)^2)^(k+1).

Original entry on oeis.org

1, 1, 1, 3, 13, 40, 177, 965, 4733, 28103, 184065, 1191888, 8713549, 67005689, 528870257, 4526024267, 40051790333, 368513578472, 3583302492545, 35868588067501, 373781214260749, 4052932682659599, 45218033687522481, 523234757502985824, 6245693941097387773
Offset: 0

Views

Author

Seiichi Manyama, Feb 20 2023

Keywords

Crossrefs

Cf. A000248, A360788.
Cf. A360782, A360592.

Programs

  • Mathematica
    Join[{1}, Table[Sum[Binomial[n-k,k] * (n-2*k)^(2*k), {k,0,n/2}], {n,1,30}]] (* Vaclav Kotesovec, Feb 21 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(k*x)^2)^(k+1)))
  • PARI
    a(n) = sum(k=0, n\2, (n-2*k)^(2*k)*binomial(n-k, k));

Formula

a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^(2*k) * binomial(n-k,k).

A360794 Expansion of Sum_{k>0} x^k / (1 - k * x^k)^(k+1).

Original entry on oeis.org

1, 3, 4, 11, 6, 43, 8, 109, 100, 281, 12, 1507, 14, 1863, 3376, 6937, 18, 26245, 20, 53211, 63022, 67739, 24, 572413, 78776, 372945, 1087048, 1761719, 30, 7362871, 32, 9947953, 16897486, 10027349, 8011116, 123101515, 38, 49807779, 241823440, 361722421, 42
Offset: 1

Views

Author

Seiichi Manyama, Feb 21 2023

Keywords

Crossrefs

Cf. A360782, A360783.
Cf. A081543, A324158.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, #^(n/# - 1) * Binomial[# + n/# - 1, #] &]; Array[a, 40] (* Amiram Eldar, Jul 31 2023 *)
  • PARI
    my(N=50, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-k*x^k)^(k+1)))
  • PARI
    a(n) = sumdiv(n, d, d^(n/d-1)*binomial(d+n/d-1, d));

Formula

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

A360834 Expansion of Sum_{k>=0} (k * x)^k / (1 - (k * x)^2)^(k+1).

Original entry on oeis.org

1, 1, 4, 29, 304, 4100, 67520, 1314167, 29520128, 751658635, 21393444864, 673046604600, 23192501108736, 868730852002205, 35145114836811776, 1527192185786650417, 70941146068492943360, 3508043437942077557884, 183989995827118805352448
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2023

Keywords

Crossrefs

Cf. A072034, A360835.
Cf. A000045, A360782, A360787.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-(k*x)^2)^(k+1)))
  • PARI
    a(n) = sum(k=0, n\2, (n-2*k)^n*binomial(n-k, k));

Formula

a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^n * binomial(n-k,k).

A360810 Expansion of Sum_{k>=0} ( x / (1 - k * x^2) )^k.

Original entry on oeis.org

1, 1, 1, 2, 5, 11, 29, 81, 229, 696, 2181, 7045, 23653, 81433, 288173, 1046814, 3887749, 14768783, 57275541, 226462801, 912443397, 3741515804, 15603500797, 66134448329, 284660214181, 1243605590897, 5511058189989, 24760003963802, 112726590916645
Offset: 0

Views

Author

Seiichi Manyama, Feb 21 2023

Keywords

Crossrefs

Cf. A324158, A360782.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (x/(1-k*x^2))^k))
  • PARI
    a(n) = sum(k=0, n\2, (n-2*k)^k*binomial(n-k-1, k));

Formula

a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^k * binomial(n-k-1,k).
Showing 1-5 of 5 results.

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