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 1-4 of 4 results.

A349962 a(n) = Sum_{k=0..n} (2*k)^k.

Original entry on oeis.org

1, 3, 19, 235, 4331, 104331, 3090315, 108503819, 4403471115, 202762761483, 10442762761483, 594761064172811, 37115108500229387, 2518267981703965963, 184577387811646500107, 14533484387811646500107, 1223459304002440821206283, 109651494909968373175414027
Offset: 0

Views

Author

Seiichi Manyama, Dec 07 2021

Keywords

Comments

Partial sums of A062971.

Links

Crossrefs

Cf. A062970, A062971, A349961, A349963, A349970.

Programs

  • Mathematica
    a[n_] := Sum[If[k == 0, 1, (2*k)^k], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Dec 07 2021 *)
  • PARI
    a(n) = sum(k=0, n, (2*k)^k);

Formula

a(n) ~ 2^n * n^n. - Vaclav Kotesovec, Dec 07 2021

A349963 a(n) = Sum_{k=0..n} (2*k)^n.

Original entry on oeis.org

1, 2, 20, 288, 5664, 141600, 4298944, 153638912, 6319260672, 294044152320, 15272286131200, 875880428003328, 54976337351106560, 3748609104907476992, 275924407293425336320, 21806398621389422592000, 1841661678145084557099008, 165530736067119754944577536
Offset: 0

Views

Author

Seiichi Manyama, Dec 07 2021

Keywords

Crossrefs

Cf. A031971, A185393, A249459, A349970.

Programs

  • Mathematica
    a[n_] := Sum[If[k == n == 0, 1, (2*k)^n], {k, 0, n}]; Array[a, 18, 0] (* Amiram Eldar, Dec 07 2021 *)
  • PARI
    a(n) = sum(k=0, n, (2*k)^n);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (2*k*x)^k/(1-2*k*x)))

Formula

G.f.: Sum_{k>=0} (2*k * x)^k/(1 - 2*k * x).
a(n) = 2^n * A031971(n).
a(n) ~ c * 2^n * n^n, where c = 1/(1 - 1/exp(1)) = A185393. - Vaclav Kotesovec, Dec 07 2021

A349969 a(n) = Sum_{k=0..n} (k*n)^(n-k).

Original entry on oeis.org

1, 1, 3, 16, 141, 1871, 34951, 873174, 27951929, 1107415549, 52891809491, 2987861887924, 196828568831365, 14950745148070499, 1296606974501951743, 127238563043551898986, 14012626653816435643633, 1719136634276882827095009, 233448782800118609096218891
Offset: 0

Views

Author

Seiichi Manyama, Dec 07 2021

Keywords

Links

Crossrefs

Cf. A026898, A155956, A349964, A349970.

Programs

  • Mathematica
    a[n_] := Sum[If[k == n == 0, 1, (k*n)^(n - k)], {k, 0, n}]; Array[a, 19, 0] (* Amiram Eldar, Dec 07 2021 *)
  • PARI
    a(n) = sum(k=0, n, (k*n)^(n-k));

Formula

a(n) = [x^n] Sum_{k>=0} x^k/(1 - n*k * x).
a(n) ~ sqrt(2*Pi/(n*(1 + LambertW(exp(1)*n^2)))) * (n^2/LambertW(exp(1)*n^2))^(n + 1/2 - n/LambertW(exp(1)*n^2)). - Vaclav Kotesovec, Dec 07 2021

A359134 a(n) = Sum_{d|n} (2*d)^(n/d - 1).

Original entry on oeis.org

1, 3, 5, 13, 17, 55, 65, 201, 293, 779, 1025, 3365, 4097, 12303, 17781, 49681, 65537, 204547, 262145, 791549, 1095429, 3145751, 4194305, 12897625, 16787217, 50331675, 68788805, 201591509, 268435457, 815505231, 1073741825, 3223326753, 4355433957, 12884901923
Offset: 1

Views

Author

Seiichi Manyama, Jan 13 2023

Keywords

Links

Crossrefs

Cf. A087909, A278741, A349970, A359733, A359796.

Programs

  • Mathematica
    a[n_] := DivisorSum[n, (2*#)^(n/# - 1) &]; Array[a, 30] (* Amiram Eldar, Aug 14 2023 *)
  • PARI
    a(n) = sumdiv(n, d, (2*d)^(n/d-1));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, x^k/(1-2*k*x^k)))

Formula

G.f.: Sum_{k>0} x^k / (1 - 2 * k * x^k).
If p is prime, a(p) = 1 + 2^(p-1).
Showing 1-4 of 4 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.