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 1-6 of 6 results.

A349880 Expansion of Sum_{k>=0} x^k/(1 - k^3 * x).

Original entry on oeis.org

1, 1, 2, 10, 93, 1307, 28002, 842196, 33388393, 1717595949, 111931584098, 8979468552886, 872315432217509, 101425775048588759, 13924209725224120770, 2229705716369149960592, 412760812611799202662609, 87644186710319273062637625, 21180850892383599137766296770
Offset: 0

Views

Author

Seiichi Manyama, Dec 03 2021

Keywords

Links

Crossrefs

Cf. A026898, A234568, A349881.
Cf. A349857.

Programs

  • PARI
    a(n, s=0, t=3) = sum(k=0, n, k^(t*(n-k)+s));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-k^3*x)))

Formula

a(n) = Sum_{k=0..n} k^(3*(n-k)).
a(n) ~ sqrt(2*Pi/3) * (n/LambertW(exp(1)*n))^(1/2 + 3*n - 3*n/LambertW(exp(1)*n)) / sqrt(1 + LambertW(exp(1)*n)). - Vaclav Kotesovec, Dec 04 2021

A349881 Expansion of Sum_{k>=0} x^k/(1 - k^4 * x).

Original entry on oeis.org

1, 1, 2, 18, 339, 10915, 663140, 61264436, 8044351557, 1536980041573, 402558463751974, 137204787854813174, 60668198155262809815, 34351266752678243067591, 24185207999807747975188552, 20842786946335533698574605528
Offset: 0

Views

Author

Seiichi Manyama, Dec 03 2021

Keywords

Comments

In general, for t>=1 and s>=0, Sum_{k=0..n} k^(t*(n-k)+s) ~ sqrt(2*Pi) * ((n + s/t)/LambertW(exp(1)*(n + s/t)))^(1/2 + (t*n + s) * (1 - 1/LambertW(exp(1)*(n + s/t)))) / sqrt(t*(1 + LambertW(exp(1)*(n + s/t)))). - Vaclav Kotesovec, Dec 04 2021

Links

Crossrefs

Cf. A026898, A234568, A349880.
Cf. A349858.

Programs

  • Mathematica
    a[n_] := Sum[If[k == n - k == 0, 1, k^(4*(n - k))], {k, 0, n}]; Array[a, 16, 0] (* Amiram Eldar, Dec 04 2021 *)
  • PARI
    a(n, s=0, t=4) = sum(k=0, n, k^(t*(n-k)+s));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-k^4*x)))

Formula

a(n) = Sum_{k=0..n} k^(4*(n-k)).
a(n) ~ sqrt(Pi/2) * (n/LambertW(exp(1)*n))^(1/2 + 4*n - 4*n/LambertW(exp(1)*n)) / sqrt(1 + LambertW(exp(1)*n)). - Vaclav Kotesovec, Dec 04 2021

A349882 Expansion of Sum_{k>=0} k^2 * x^k/(1 - k^2 * x).

Original entry on oeis.org

0, 1, 5, 26, 162, 1267, 12343, 145652, 2036148, 33192789, 622384729, 13263528350, 318121600694, 8517247764135, 252725694989611, 8258153081400856, 295515712276222952, 11523986940937975401, 487562536078882116717, 22291094729329088403298
Offset: 0

Views

Author

Seiichi Manyama, Dec 03 2021

Keywords

Links

Crossrefs

Cf. A003101, A234568, A249459, A349863, A359659.

Programs

  • Mathematica
    a[n_] := Sum[If[k == n - k + 1 == 0, 1, k^(2*(n - k + 1))], {k, 0, n}]; Array[a, 20, 0] (* Amiram Eldar, Dec 04 2021 *)
  • PARI
    a(n, s=2, t=2) = sum(k=0, n, k^(t*(n-k)+s));
  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, k^2*x^k/(1-k^2*x))))
  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(sum(k=1, N, x^k/(1-(k+1)^2*x)))) \\ Seiichi Manyama, Jan 12 2023

Formula

a(n) = Sum_{k=0..n} k^(2*(n-k+1)).
a(n) = A234568(n+1) - 1. - Hugo Pfoertner, Dec 04 2021
a(n) ~ sqrt(Pi) * ((n+1)/LambertW(exp(1)*(n+1)))^(5/2 + 2*n - 2*(n+1)/LambertW(exp(1)*(n+1))) / sqrt(1 + LambertW(exp(1)*(n+1))). - Vaclav Kotesovec, Dec 04 2021
G.f.: Sum_{k>=1} x^k/(1 - (k+1)^2 * x). - Seiichi Manyama, Jan 12 2023

A355471 Expansion of Sum_{k>=0} (x/(1 - k^2 * x))^k.

Original entry on oeis.org

1, 1, 2, 10, 77, 808, 11257, 196072, 4136897, 103755904, 3034193921, 101901347944, 3885951145969, 166605168800704, 7961498177012993, 420976047757358776, 24475992585921169553, 1556007778666449968128, 107625967130820901112833
Offset: 0

Views

Author

Seiichi Manyama, Jul 03 2022

Keywords

Crossrefs

Cf. A080108, A135746, A234568, A355463, A355472.

Programs

  • Mathematica
    Flatten[{1, Table[Sum[Binomial[n-1,k-1] * k^(2*(n-k)), {k,1,n}], {n,1,20}]}] (* Vaclav Kotesovec, Feb 16 2023 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (x/(1-k^2*x))^k))
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, k^(2*(n-k))*binomial(n-1, k-1)));

Formula

a(n) = Sum_{k=1..n} k^(2*(n-k)) * binomial(n-1,k-1) for n > 0.

A356672 a(n) = n! * Sum_{k=0..n} k^(2*(n-k))/k!.

Original entry on oeis.org

1, 1, 3, 19, 253, 5661, 188191, 8983423, 594848409, 52174034713, 5852229698971, 822684190381131, 142739480367287893, 30074750245383836149, 7575373641076070706423, 2252600759590927171373431, 783103569459739402827046321, 315587346190678252431713684913
Offset: 0

Views

Author

Seiichi Manyama, Aug 22 2022

Keywords

Crossrefs

Cf. A354436, A356673.
Cf. A234568, A356628.

Programs

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

Formula

E.g.f: Sum_{k>=0} x^k / (k! * (1 - k^2 * x)).

A359660 a(n) = Sum_{k=0..n} k^(2 * (n-k) + 1).

Original entry on oeis.org

0, 1, 3, 12, 64, 441, 3855, 41464, 533736, 8071785, 141351715, 2829417276, 64038928728, 1624347614737, 45822087138879, 1427872211276376, 48858282302548240, 1826209988254883889, 74216973833968292451, 3265676709281560408780
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2023

Keywords

Crossrefs

Cf. A234568, A349882.
Cf. A000217, A003101.

Programs

  • PARI
    a(n, s=1, t=2) = sum(k=0, n, k^(t*(n-k)+s));
  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, k*x^k/(1-k^2*x))))

Formula

G.f.: Sum_{k>=0} k * x^k/(1 - k^2 * x).
Showing 1-6 of 6 results.

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