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 1-4 of 4 results.

A356408 Expansion of e.g.f. ( Product_{k>0} 1/(1 - x^k/k) )^(1/(1-x)).

Original entry on oeis.org

1, 1, 5, 29, 216, 1919, 20012, 236977, 3145832, 46122546, 739703182, 12865212172, 241040899668, 4836265824740, 103410589256452, 2346358252787094, 56285005757022752, 1422783492250963296, 37790069818311971640, 1051924374853915254048
Offset: 0

Views

Author

Seiichi Manyama, Aug 05 2022

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-x^k/k)^(1/(1-x))))
    
  • PARI
    a356406(n) = n!*sum(k=1, n, sumdiv(k, d, 1/(d*(k/d)^d)));
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a356406(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A356406(k) * binomial(n-1,k-1) * a(n-k).

A356407 a(n) = n! * Sum_{k=1..n} Sum_{d|k} 1/(d * ((k/d)!)^d).

Original entry on oeis.org

1, 4, 15, 70, 375, 2411, 17598, 146490, 1359291, 13978597, 157393368, 1929989029, 25568858978, 364288345409, 5551537358188, 90142504077194, 1553345359200299, 28317316174307405, 544431381017568696, 11010510372888267555, 233653645911730002976
Offset: 1

Views

Author

Seiichi Manyama, Aug 05 2022

Keywords

Crossrefs

Programs

  • PARI
    a(n) = n!*sum(k=1, n, sumdiv(k, d, 1/(d*(k/d)!^d)));
    
  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(-sum(k=1, N, log(1-x^k/k!))/(1-x)))

Formula

E.g.f.: -(1/(1-x)) * Sum_{k>0} log(1 - x^k/k!).
a(n) = n! * Sum_{k=1..n} A182926(k)/k!.

A356579 Expansion of e.g.f. ( Product_{k>0} 1/(1 - x^k/k!) )^x.

Original entry on oeis.org

1, 0, 2, 6, 24, 170, 990, 8267, 67928, 661698, 6923010, 78997457, 983728812, 13101433501, 187893745130, 2869108871085, 46643882262448, 803224515183482, 14618310020427402, 280340253237270977, 5651276469430635620, 119483759770082806035, 2644015844432596590946
Offset: 0

Views

Author

Seiichi Manyama, Aug 12 2022

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-x^k/k!)^x))
    
  • PARI
    a182926(n) = n!*sumdiv(n, d, 1/(d*(n/d)!^d));
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=2, i, j*a182926(j-1)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1, a(1) = 0; a(n) = Sum_{k=2..n} k * A182926(k-1) * binomial(n-1,k-1) * a(n-k).

A356599 Expansion of e.g.f. ( Product_{k>0} 1/(1 - x^k/k!) )^exp(x).

Original entry on oeis.org

1, 1, 5, 25, 159, 1201, 10488, 102901, 1121375, 13406353, 174284898, 2445111373, 36799134584, 591042564425, 10086822013726, 182218681622851, 3472980343846199, 69632877583186121, 1464890891351327598, 32260213678562913097, 742152913359395190170
Offset: 0

Views

Author

Seiichi Manyama, Aug 15 2022

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(1/prod(k=1, N, 1-x^k/k!)^exp(x)))
    
  • PARI
    a354341(n) = n!*sum(k=1, n, sumdiv(k, d, 1/(d*(k/d)!^d))/(n-k)!);
    a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354341(j)*binomial(i-1, j-1)*v[i-j+1])); v;

Formula

a(0) = 1; a(n) = Sum_{k=1..n} A354341(k) * binomial(n-1,k-1) * a(n-k).
Showing 1-4 of 4 results.