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 1-7 of 7 results.

A355463 Expansion of Sum_{k>=0} (x/(1 - k^k * x))^k.

Original entry on oeis.org

1, 1, 2, 10, 131, 5656, 869097, 490286392, 1264458639313, 12443651667592768, 681538604797281047489, 153070077563816488157872384, 205935348854901274982393017521537, 1352544986573612111579941739713633174912
Offset: 0

Views

Author

Seiichi Manyama, Jul 03 2022

Keywords

Links

Crossrefs

Cf. A193198, A193199, A349893, A355464.
Cf. A080108, A355471, A355472.

Programs

  • Mathematica
    Flatten[{1, Table[Sum[Binomial[n-1,k-1] * k^(k*(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^k*x))^k))
  • PARI
    a(n) = if(n==0, 1, sum(k=1, n, k^(k*(n-k))*binomial(n-1, k-1)));

Formula

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

A355464 Expansion of Sum_{k>=0} x^k/(1 - k^k * x)^(k+1).

Original entry on oeis.org

1, 2, 4, 17, 210, 9217, 1399298, 811229225, 2071392232962, 20710319937493889, 1137259214532706572162, 255141201504146525745627265, 348787971214016591166179037803522, 2262996819897931095524655885144485185409
Offset: 0

Views

Author

Seiichi Manyama, Jul 03 2022

Keywords

Crossrefs

Cf. A086331, A320287, A349893, A355440, A355463.
Cf. A000248, A135746, A355473.

Programs

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

Formula

E.g.f.: Sum_{k>=0} exp(k^k * x) * x^k/k!.
a(n) = Sum_{k=0..n} k^(k*(n-k)) * binomial(n,k).

A356674 a(n) = n! * Sum_{k=0..n} k^(k*(n-k))/k!.

Original entry on oeis.org

1, 2, 5, 25, 349, 19941, 4440391, 4382699203, 17687865017481, 356274213630958297, 33338407933090938442411, 16214021627369697901867402911, 43817834057167927861655409052462093, 595284492835035398061242850538179192931525
Offset: 0

Views

Author

Seiichi Manyama, Aug 22 2022

Keywords

Links

Crossrefs

Cf. A354436, A356672, A356673.
Cf. A327578, A349893.

Programs

  • Mathematica
    Table[n!*(1 + Sum[k^(k*(n-k))/k!, {k, 1, n}]), {n, 0, 12}] (* Vaclav Kotesovec, Nov 27 2022 *)
  • PARI
    a(n) = n!*sum(k=0, n, k^(k*(n-k))/k!);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(k!*(1-k^k*x)))))

Formula

E.g.f: Sum_{k>=0} x^k / (k! * (1 - k^k * x)).
log(a(n)) ~ n^2*log(n)/4 * (1 - log(2)/log(n) + 1/(4*log(n)^2)). - Vaclav Kotesovec, Nov 27 2022

A359659 a(n) = Sum_{k=0..n} k^(k * (n-k+1)).

Original entry on oeis.org

1, 2, 6, 45, 1051, 88602, 27121964, 37004504305, 198705527223757, 5595513387083114570, 686714367475480207331582, 468422339816915120237104999421, 1664212116512828935888786624225704855
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2023

Keywords

Links

Crossrefs

Cf. A026898, A349893, A359658.
Cf. A031971, A349836, A349883.
Cf. A003101, A349882.

Programs

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

Formula

G.f.: Sum_{k>=0} (k * x)^k/(1 - k^k * x).
G.f.: Sum_{k>=0} x^k/(1 - (k+1)^(k+1) * x).
a(n) = A349893(n+1) - 1.

A349894 a(n) = Sum_{k=0..n} (-1)^(n-k) * k^(k*(n-k)).

Original entry on oeis.org

1, 0, 1, -3, -10, 410, 42985, -6527829, -24060996846, -6613442955828, 3882375189467092921, 235121650953066124724477, -289337164954511885810252000250, -995208334663809003504695464745010282, 13325880481925983143500510271865447222057073
Offset: 0

Views

Author

Seiichi Manyama, Dec 04 2021

Keywords

Links

Crossrefs

Cf. A349856, A349857, A349858, A349891, A349893.

Programs

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

Formula

G.f.: Sum_{k>=0} x^k/(1 + k^k * x).

A358688 a(n) = n! * Sum_{k=0..n} k^(k * (n-k)) / (n-k)!.

Original entry on oeis.org

1, 2, 5, 34, 869, 75866, 28213327, 39049033346, 256215628707257, 7710689746589777938, 1063776147486867074877851, 870059224717752809087935599002, 3104894940194751778363241199111802885, 77521065749331962430758061530260243383954602
Offset: 0

Views

Author

Seiichi Manyama, Nov 26 2022

Keywords

Links

Crossrefs

Cf. A006153, A193421, A358687.
Cf. A349893, A356674.

Programs

  • Mathematica
    Table[1 + n!*Sum[k^(k*(n-k))/(n-k)!, {k, 1, n}], {n, 0, 12}] (* Vaclav Kotesovec, Nov 27 2022 *)
  • PARI
    a(n) = n!*sum(k=0, n, k^(k*(n-k))/(n-k)!);
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k*exp(x)^k^k)))
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, k!*x^k/(1-k^k*x)^(k+1)))

Formula

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

A359658 a(n) = Sum_{k=0..n} k^(k * (n-k) + 1).

Original entry on oeis.org

0, 1, 3, 12, 118, 3345, 337337, 117813304, 182877273548, 1095343802746641, 33833602932485958015, 4588786457956655542361532, 3347980595386754115503487966082, 13023291362471615806961306534915589217
Offset: 0

Views

Author

Seiichi Manyama, Jan 10 2023

Keywords

Crossrefs

Cf. A349893, A359659.

Programs

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

Formula

G.f.: Sum_{k>=0} k * x^k/(1 - k^k * x).
Showing 1-7 of 7 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.