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.

A360782 Expansion of Sum_{k>=0} x^k / (1 - k*x^2)^(k+1).

Original entry on oeis.org

1, 1, 1, 3, 7, 16, 45, 125, 363, 1127, 3561, 11696, 39727, 138113, 494213, 1811075, 6784115, 25985928, 101520833, 404305549, 1640002039, 6767576175, 28395916893, 121048681024, 523902418555, 2300906314849, 10248029334297, 46266088140291
Offset: 0

Views

Author

Seiichi Manyama, Feb 20 2023

Keywords

Crossrefs

Cf. A000248, A360783.
Cf. A000045, A360592.

Programs

  • Mathematica
    Join[{1}, Table[Sum[Binomial[n-k,k] * (n-2*k)^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)^k*binomial(n-k, k));

Formula

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

A360788 Expansion of Sum_{k>=0} x^k / (1 - (k*x)^3)^(k+1).

Original entry on oeis.org

1, 1, 1, 1, 3, 25, 109, 324, 1135, 8803, 64189, 337854, 1707319, 13421410, 121248893, 894378619, 6082868725, 53046554917, 543432115477, 4989423130739, 42565774604131, 421544374075072, 4781440892689533, 51342685464272591, 522295380717090265
Offset: 0

Views

Author

Seiichi Manyama, Feb 20 2023

Keywords

Crossrefs

Cf. A000248, A360787.
Cf. A360783.

Programs

  • PARI
    my(N=30, 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)^(3*k)*binomial(n-2*k, k));

Formula

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

A360811 Expansion of Sum_{k>=0} ( x / (1 - k * x^3) )^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 10, 18, 38, 91, 211, 472, 1108, 2754, 6881, 17101, 43443, 113565, 300142, 797191, 2147414, 5883976, 16293712, 45471429, 128285353, 366266188, 1055534118, 3066483484, 8989837397, 26602652605, 79370560477, 238606427241, 722973445270
Offset: 0

Views

Author

Seiichi Manyama, Feb 21 2023

Keywords

Links

Crossrefs

Cf. A324158, A360783.

Programs

  • Maple
    N:= 100:
F:= 1 + add((x/(1-k*x^3))^k, k=1..N):
S:= series(F,x,N+1):
seq(coeff(S,x,k),k=0..N); # Robert Israel, Feb 21 2024
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, (x/(1-k*x^3))^k))
  • PARI
    a(n) = sum(k=0, n\3, (n-3*k)^k*binomial(n-2*k-1, k));

Formula

a(n) = Sum_{k=0..floor(n/3)} (n-3*k)^k * binomial(n-2*k-1,k).

A360835 Expansion of Sum_{k>=0} (k * x)^k / (1 - (k * x)^3)^(k+1).

Original entry on oeis.org

1, 1, 4, 27, 258, 3221, 49572, 905466, 19122502, 458161191, 12275530636, 363646493044, 11801356347294, 416365459777150, 15867258718677348, 649548679156603983, 28426564854590132236, 1324406974148881529057, 65448443631801436742052
Offset: 0

Views

Author

Seiichi Manyama, Feb 22 2023

Keywords

Crossrefs

Cf. A072034, A360834.
Cf. A000930, A360783, A360788.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-(k*x)^3)^(k+1)))
  • PARI
    a(n) = sum(k=0, n\3, (n-3*k)^n*binomial(n-2*k, k));

Formula

a(n) = Sum_{k=0..floor(n/3)} (n-3*k)^n * binomial(n-2*k,k).
Showing 1-5 of 5 results.

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

Content is available under The OEIS End-User License Agreement.