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.

A294392 E.g.f.: exp(Sum_{n>=1} A001227(n) * x^n).

Original entry on oeis.org

1, 1, 3, 19, 97, 801, 7411, 73123, 821409, 10977697, 151612291, 2286137811, 38308830913, 669163118209, 12649211055027, 257559356068771, 5432325991339201, 121949878889492673, 2907330680764076419, 71860237654425159187, 1871308081194213959841
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2017

Keywords

Crossrefs

E.g.f.: exp(Sum_{n>=1} (Sum_{d|n and d is odd} d^k) * x^n): this sequence (k=0), A294394 (k=1), A294395 (k=2).

Programs

  • Mathematica
    a[n_] := a[n] = If[n == 0, 1, Sum[k*DivisorSum[k, Mod[#, 2] &]*a[n - k], {k, 1, n}]/n]; Table[n!*a[n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 07 2018 *)
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, sumdiv(k, d, d%2)*x^k))))

Formula

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A001227(k)*a(n-k)/(n-k)! for n > 0.
E.g.f.: Product_{k>=1} exp(x^(2*k-1)/(1 - x^(2*k-1))). - Ilya Gutkovskiy, Nov 27 2017
Conjecture: log(a(n)/n!) ~ sqrt(n*log(n)). - Vaclav Kotesovec, Sep 07 2018

A294394 E.g.f.: exp(Sum_{n>=1} A000593(n) * x^n).

Original entry on oeis.org

1, 1, 3, 31, 145, 1641, 17731, 194503, 2676801, 40644145, 667689571, 11514903951, 227665389073, 4578990563161, 100913115588195, 2372334731747191, 57930324367791361, 1509398686720812513, 41341036374519788611, 1184009909077133031295
Offset: 0

Views

Author

Seiichi Manyama, Oct 30 2017

Keywords

Crossrefs

E.g.f.: exp(Sum_{n>=1} (Sum_{d|n and d is odd} d^k) * x^n): A294392 (k=0), this sequence (k=1), A294395 (k=2).

Programs

  • Mathematica
    a[n_] := a[n] = If[n == 0, 1, Sum[k*Sum[-(-1)^d*k/d, {d, Divisors[k]}]*a[n - k], {k, 1, n}]/n]; Table[n!*a[n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 07 2018 *)
  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, sumdiv(k, d, d*(d%2))*x^k))))

Formula

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000593(k)*a(n-k)/(n-k)! for n > 0.
a(n) ~ Pi^(1/3) * exp((3*Pi)^(2/3) * n^(2/3) / 2^(4/3) - 1/24 - n) * n^(n - 1/6) / (2^(1/6) * 3^(2/3)). - Vaclav Kotesovec, Sep 07 2018

A294461 E.g.f.: exp(-Sum_{n>=1} A050999(n) * x^n).

Original entry on oeis.org

1, -1, -1, -55, 217, -2441, 41911, -343519, 10531025, -123024817, 2722259791, -64395229031, 1218005521129, -36874422541945, 785879799954887, -25331247487596751, 708096286059632161, -21422225147712360929, 741754828422824400415
Offset: 0

Views

Author

Seiichi Manyama, Oct 31 2017

Keywords

Crossrefs

E.g.f.: exp(-Sum_{n>=1} (Sum_{d|n and d is odd} d^k) * x^n): A294459 (k=0), A294460 (k=1), this sequence (k=2).

Programs

  • PARI
    N=66; x='x+O('x^N); Vec(serlaplace(exp(-sum(k=1, N, sumdiv(k, d, d^2*(d%2))*x^k))))

Formula

a(0) = 1 and a(n) = (-1) * (n-1)! * Sum_{k=1..n} k*A050999(k)*a(n-k)/(n-k)! for n > 0.
Showing 1-3 of 3 results.