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.

A346395 Expansion of e.g.f. -log(1 - x) * exp(3*x).

Original entry on oeis.org

0, 1, 7, 38, 192, 969, 5115, 29322, 187992, 1370745, 11392839, 107043606, 1122823944, 12989320785, 164040593067, 2243143392138, 32994768719376, 519229765892241, 8701862242296807, 154700700117472422, 2907409255935736752, 57588370882960384377, 1198954118077558162875
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 15 2021

Keywords

Links

Crossrefs

Cf. A002104, A053486, A346394, A346396.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[-Log[1 - x] Exp[3 x], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[3^k/((n - k) k!), {k, 0, n - 1}], {n, 0, 22}]
  • PARI
    a_vector(n) = my(v=vector(n+1, i, if(i==2, 1, 0))); for(i=2, n, v[i+1]=(i+2)*v[i]-3*(i-1)*v[i-1]+3^(i-1)); v; \\ Seiichi Manyama, May 27 2022

Formula

a(n) = n! * Sum_{k=0..n-1} 3^k / ((n-k) * k!).
a(n) ~ exp(3) * (n-1)!. - Vaclav Kotesovec, Aug 09 2021
a(0) = 0, a(1) = 1, a(n) = (n+2) * a(n-1) - 3 * (n-1) * a(n-2) + 3^(n-1). - Seiichi Manyama, May 27 2022

A346394 Expansion of e.g.f. -log(1 - x) * exp(2*x).

Original entry on oeis.org

0, 1, 5, 20, 78, 324, 1520, 8336, 53872, 405600, 3492416, 33798016, 362543104, 4264455168, 54540715008, 753246711808, 11168972683264, 176937613586432, 2982069587042304, 53271637651996672, 1005385746384846848, 19987620914387812352, 417489079682758213632
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 15 2021

Keywords

Links

Crossrefs

Cf. A002104, A010842, A066534, A346395, A346396.

Programs

  • Mathematica
    nmax = 22; CoefficientList[Series[-Log[1 - x] Exp[2 x], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Sum[2^k/((n - k) k!), {k, 0, n - 1}], {n, 0, 22}]
  • PARI
    a_vector(n) = my(v=vector(n+1, i, if(i==2, 1, 0))); for(i=2, n, v[i+1]=(i+1)*v[i]-2*(i-1)*v[i-1]+2^(i-1)); v; \\ Seiichi Manyama, May 27 2022

Formula

a(n) = n! * Sum_{k=0..n-1} 2^k / ((n-k) * k!).
a(n) = Sum_{k=0..n} binomial(n,k) * A002104(k).
a(n) ~ exp(2) * (n-1)!. - Vaclav Kotesovec, Aug 09 2021
a(0) = 0, a(1) = 1, a(n) = (n+1) * a(n-1) - 2 * (n-1) * a(n-2) + 2^(n-1). - Seiichi Manyama, May 27 2022

A353548 Expansion of e.g.f. -log(1-4*x) * exp(x)/4.

Original entry on oeis.org

0, 1, 6, 47, 540, 8429, 166210, 3952955, 109981816, 3502905369, 125648153278, 5011458069639, 219987094389524, 10538817637744005, 547118005892177018, 30595552548140425747, 1833501625083035349488, 117219490267316310468913
Offset: 0

Views

Author

Seiichi Manyama, May 27 2022

Keywords

Crossrefs

Cf. A002104, A353546, A353547, A353549.
Cf. A346396.
Essentially partial sums of A056545.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); concat(0, Vec(serlaplace(-log(1-4*x)*exp(x)/4)))
  • PARI
    a(n) = n!*sum(k=0, n-1, 4^(n-1-k)/((n-k)*k!));
  • PARI
    a_vector(n) = my(v=vector(n+1, i, if(i==2, 1, 0))); for(i=2, n, v[i+1]=(4*i-3)*v[i]-4*(i-1)*v[i-1]+1); v;

Formula

a(n) = n! * Sum_{k=0..n-1} 4^(n-1-k) / ((n-k) * k!).
a(0) = 0, a(1) = 1, a(n) = (4 * n - 3) * a(n-1) - 4 * (n-1) * a(n-2) + 1.
a(n) ~ (n-1)! * exp(1/4) * 4^(n-1). - Vaclav Kotesovec, Jun 08 2022
Showing 1-3 of 3 results.

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