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 1-3 of 3 results.

A360699 G.f.: Sum_{k>=0} (1 + k*x)^k * x^(2*k).

Original entry on oeis.org

1, 0, 1, 1, 1, 4, 5, 9, 28, 43, 97, 281, 507, 1286, 3666, 7494, 20470, 58725, 132484, 381700, 1113180, 2719887, 8171219, 24337511, 63524916, 197606643, 602261524, 1662206380, 5328738685, 16628469912, 48148703533, 158544768073, 506473892417, 1529218062752, 5159071807165
Offset: 0

Views

Author

Vaclav Kotesovec, Feb 16 2023

Keywords

Links

Crossrefs

Cf. A360592, A360707, A360479.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Sum[(1 + k*x)^k * x^(2*k), {k, 0, nmax}], {x, 0, nmax}], x]
Join[{1}, Table[Sum[Binomial[k, n - 2*k] * k^(n - 2*k), {k, 0, n}], {n, 1, 40}]]

Formula

a(n) = Sum_{k=0..n} binomial(k,n-2*k) * k^(n-2*k).
log(a(n)) ~ n/3 * log(n/3).
a(n) ~ exp(exp(1/3)*n^(1/3)/3^(1/3)) * n^(n/3) / 3^(n/3 + 1) * (1 + (3^(1/3)/(8*exp(1/3)) - 4*exp(2/3)/3^(5/3)) / n^(1/3) + (67/(128*3^(1/3)*exp(2/3)) + 8*exp(4/3)/3^(10/3)) / n^(2/3)).

A360747 Expansion of Sum_{k>=0} (x * (1 + (k * x)^3))^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 17, 82, 257, 690, 3484, 26978, 160347, 726085, 3529206, 26885924, 220706533, 1474182023, 8834370165, 65392181686, 604821608674, 5230627589958, 39543579302104, 312733691925723, 3013530105191283, 30474809255061289
Offset: 0

Views

Author

Seiichi Manyama, Feb 19 2023

Keywords

Links

Crossrefs

Cf. A360479, A360592, A360707.

Programs

  • Mathematica
    Join[{1},Table[Sum[Binomial[n - 3*k,k] * (n - 3*k)^(3*k), {k,0,n/4}], {n,1,30}]] (* Vaclav Kotesovec, Feb 19 2023 *)
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (x*(1+(k*x)^3))^k))
  • PARI
    a(n) = sum(k=0, n\4, (n-3*k)^(3*k)*binomial(n-3*k, k));

Formula

a(n) = Sum_{k=0..floor(n/4)} (n-3*k)^(3*k) * binomial(n-3*k,k).
a(n) ~ exp(exp(9/4)*n^(1/4)/sqrt(2)) * n^(3*n/4) / 2^(3*n/2 + 2) * (1 + 1/(4*sqrt(2)*exp(9/4) * n^(1/4)) + (67/(192*exp(9/2)) - 37*exp(9/2)/16) / sqrt(n) + (497/(768*sqrt(2)*exp(27/4)) - 205*exp(9/4)/(64*sqrt(2))) / n^(3/4) + (10721/3072 + 218831/(368640*exp(9)) + (1369*exp(9))/512)/n), see graph for more minor asymptotic terms. - Vaclav Kotesovec, Feb 20 2023

A360748 Expansion of Sum_{k>=0} (x * (1 + k*x^2))^k.

Original entry on oeis.org

1, 1, 1, 2, 5, 10, 21, 53, 133, 327, 861, 2361, 6469, 18168, 52757, 155221, 463077, 1412656, 4379917, 13747504, 43834213, 141866555, 464650309, 1541008295, 5176660997, 17586913779, 60400627453, 209746820056, 735953607173, 2607716976945, 9330605338485
Offset: 0

Views

Author

Seiichi Manyama, Feb 19 2023

Keywords

Links

Crossrefs

Cf. A360592, A360749.
Cf. A000930, A360730, A360479.

Programs

  • Mathematica
    Join[{1},Table[Sum[Binomial[n - 2*k,k] * (n - 2*k)^k, {k,0,n/3}], {n,1,30}]] (* Vaclav Kotesovec, Feb 20 2023 *)
  • PARI
    my(N=40, x='x+O('x^N)); Vec(sum(k=0, N, (x*(1+k*x^2))^k))
  • PARI
    a(n) = sum(k=0, n\3, (n-2*k)^k*binomial(n-2*k, k));

Formula

a(n) = Sum_{k=0..floor(n/3)} (n-2*k)^k * binomial(n-2*k,k).
a(n) ~ exp(exp(2/3)*n^(2/3)/3^(2/3) - 5*exp(4/3)*n^(1/3)/(18*3^(1/3)) + 22*exp(2)/81) * n^(n/3) / 3^(n/3 + 1) * (1 + (2*exp(2/3)/3^(5/3) - 3295*exp(8/3)/(2916*3^(2/3)))/n^(1/3) + (3^(2/3)/(8*exp(2/3)) + 35*exp(4/3)/(36*3^(1/3)) + 27379*exp(10/3)/(17496*3^(1/3)) + 10857025*exp(16/3)/(51018336*3^(1/3)))/n^(2/3)). - Vaclav Kotesovec, Feb 20 2023
Showing 1-3 of 3 results.

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