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.

A104872 Diagonal sums of A004248.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 6, 12, 27, 64, 163, 441, 1268, 3855, 12344, 41464, 145653, 533736, 2036149, 8071785, 33192790, 141351715, 622384730, 2829417276, 13263528351, 64038928728, 318121600695, 1624347614737, 8517247764136, 45822087138879
Offset: 0

Views

Author

Paul Barry, Mar 28 2005

Keywords

Links

Crossrefs

Cf. A004248, A026898, A352944.

Programs

  • PARI
    a(n) = sum(k=0, n\2, k^(n-2*k)); \\ Seiichi Manyama, Apr 09 2022
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^(2*k)/(1-k*x))) \\ Seiichi Manyama, Apr 09 2022

Formula

a(n) = Sum_{k=0..floor(n/2)} k^(n-2*k).
G.f.: Sum_{k>=0} x^(2*k) / (1 - k * x). - Seiichi Manyama, Apr 09 2022
a(n) ~ sqrt(Pi) * (n/(2*LambertW(exp(1)*n/2)))^(n + 1/2 - n/LambertW(exp(1)*n/2)) / sqrt(1 + LambertW(exp(1)*n/2)). - Vaclav Kotesovec, Apr 14 2022

A356632 a(n) = n! * Sum_{k=0..floor(n/2)} (n - 2*k)^k/2^k.

Original entry on oeis.org

1, 1, 2, 9, 48, 330, 2880, 29610, 362880, 5148360, 83462400, 1535549400, 31614105600, 724183059600, 18307441152000, 507367438578000, 15336404987904000, 502812808754256000, 17805001275629568000, 678167395781763888000, 27681559049033809920000
Offset: 0

Views

Author

Seiichi Manyama, Aug 18 2022

Keywords

Crossrefs

Cf. A356633, A356634.
Cf. A352944.

Programs

  • Mathematica
    a[n_] := n! * Sum[(n - 2*k)^k/2^k, {k, 0, Floor[n/2]}]; a[0] = 1; Array[a, 21, 0] (* Amiram Eldar, Aug 19 2022 *)
  • PARI
    a(n) = n!*sum(k=0, n\2, (n-2*k)^k/2^k);
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N, x^k/(1-k*x^2/2))))

Formula

E.g.f.: Sum_{k>=0} x^k / (1 - k*x^2/2).
a(n) ~ Pi * exp((1/LambertW(exp(1)*n/2) - 3)*n/2) * n^(3*n/2 + 1) / (sqrt(1 + LambertW(exp(1)*n/2)) * 2^((n-1)/2) * LambertW(exp(1)*n/2)^((n+1)/2)). - Vaclav Kotesovec, Nov 01 2022

A352946 a(n) = Sum_{k=0..floor(n/3)} (n-3*k)^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 6, 10, 16, 25, 42, 73, 125, 217, 391, 714, 1305, 2428, 4612, 8830, 17038, 33377, 66216, 132349, 267075, 545329, 1123693, 2333278, 4889751, 10342468, 22043954, 47340802, 102504532, 223654713, 491393646, 1087353601, 2423448817, 5437568233
Offset: 0

Views

Author

Seiichi Manyama, Apr 09 2022

Keywords

Crossrefs

Cf. A026898, A352944.
Cf. A001840.

Programs

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

Formula

G.f.: Sum_{k>=0} x^k / (1 - k * x^3).
a(n) ~ sqrt(2*Pi/3) * (n/LambertW(exp(1)*n))^(n*(1 - 1/LambertW(exp(1)*n))/3 + 1/2) / sqrt(1 + LambertW(exp(1)*n)). - Vaclav Kotesovec, Apr 14 2022

A353013 a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^(n-k).

Original entry on oeis.org

1, 1, 4, 28, 264, 3207, 47696, 839412, 17061280, 393264145, 10135913792, 288839201432, 9017184333440, 306045200463519, 11220008681600256, 441866073895351128, 18603606156815235584, 833860238440653331505, 39643749441387211150336
Offset: 0

Views

Author

Seiichi Manyama, Apr 16 2022

Keywords

Crossrefs

Cf. A031971, A353014.
Cf. A352981, A352944.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[(n - 2*k)^(n - k), {k, 0, Floor[n/2]}]; Array[a, 20, 0] (* Amiram Eldar, Apr 16 2022 *)
  • PARI
    a(n) = sum(k=0, n\2, (n-2*k)^(n-k));
  • PARI
    my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (k*x)^k/(1-k*x^2)))

Formula

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

A353016 a(n) = Sum_{k=0..floor(n/2)} (n-2*k)^(2*k).

Original entry on oeis.org

1, 1, 1, 2, 5, 11, 33, 108, 357, 1405, 5713, 24670, 117413, 574007, 3004577, 16608120, 95057925, 576245913, 3622049809, 23693870554, 161816447365, 1140392550275, 8351286979745, 63206781102116, 493344133444389, 3980464191557205, 33029872125113937, 282290255465835382
Offset: 0

Views

Author

Seiichi Manyama, Apr 16 2022

Keywords

Crossrefs

Cf. A062811, A352082, A352944.

Programs

  • Mathematica
    a[0] = 1; a[n_] := Sum[(n-2*k)^(2*k), {k, 0, Floor[n/2]}]; Array[a, 30, 0] (* Amiram Eldar, Apr 16 2022 *)
  • PARI
    a(n) = sum(k=0, n\2, (n-2*k)^(2*k));
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, x^k/(1-(k*x)^2)))

Formula

G.f.: Sum_{k>=0} x^k / (1 - (k * x)^2).
a(n) = (A062811(n) + 1)/2 for n > 0. - Hugo Pfoertner, Apr 16 2022
Showing 1-5 of 5 results.

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