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.

A351767 Expansion of e.g.f. exp( x/(1-x)^3 ) / (1-x)^3.

Original entry on oeis.org

1, 4, 25, 214, 2293, 29176, 427189, 7049890, 129178249, 2597880268, 56815155121, 1341068392654, 33951269718205, 917020113259264, 26305693331946253, 798293630021120986, 25540244079135784849, 858854698277997113620, 30274382852181639467209
Offset: 0

Views

Author

Seiichi Manyama, Mar 18 2023

Keywords

Links

Crossrefs

Column k=3 of A361616.
Cf. A052852, A361599.

Programs

  • Mathematica
    Table[n!*Sum[Binomial[n + 2*k + 2, n - k]/k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 25 2023 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x)^3)/(1-x)^3))
  • PARI
    a(n) = n! * sum(k=0, n, binomial(n+2*k+2,n-k)/k!); \\ Winston de Greef, Mar 18 2023

Formula

a(n) = n! * Sum_{k=0..n} binomial(n+2*k+2,n-k)/k! = Sum_{k=0..n} (n+2*k+2)!/(3*k+2)! * binomial(n,k).
From Vaclav Kotesovec, Mar 25 2023: (Start)
a(n) = 4*n*a(n-1) - (n-1)*(6*n - 5)*a(n-2) + (n-2)*(n-1)*(4*n - 3)*a(n-3) - (n-3)*(n-2)*(n-1)^2*a(n-4).
a(n) ~ exp(-1/27 - 3^(-5/4)*n^(1/4)/8 + sqrt(n/3)/2 + 4*3^(-3/4)*n^(3/4) - n) * n^(n + 5/8) / (2 * 3^(5/8)) * (1 + 91837/69120 * 3^(1/4)/n^(1/4)). (End)

A343884 Expansion of e.g.f. exp( x/(1-x)^2 ) / (1-x)^2.

Original entry on oeis.org

1, 3, 15, 103, 885, 9051, 106843, 1425495, 21166953, 345678355, 6150501831, 118313311623, 2444917863325, 53982840948843, 1267645359117075, 31531781398100791, 827910838693667793, 22874802838645217955, 663243613324249850623, 20130710499843811837095
Offset: 0

Views

Author

Seiichi Manyama, Mar 18 2023

Keywords

Links

Crossrefs

Column k=2 of A361616.
Cf. A000262, A361598.

Programs

  • Mathematica
    Table[n!*Sum[Binomial[n + k + 1, n - k]/k!, {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 25 2023 *)
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-x)^2)/(1-x)^2))
  • PARI
    a(n) = n! * sum(k=0, n, binomial(n+k+1,n-k)/k!) \\ Winston de Greef, Mar 19 2023

Formula

a(n) = n! * Sum_{k=0..n} binomial(n+k+1,n-k)/k! = Sum_{k=0..n} (n+k+1)!/(2*k+1)! * binomial(n,k).
From Vaclav Kotesovec, Mar 25 2023: (Start)
a(n) ~ exp(-1/12 + 3*2^(-2/3)*n^(2/3) - n) * n^(n + 1/2) / sqrt(6) * (1 + 2^(1/3)/n^(1/3) + 323/(360*2^(1/3)*n^(2/3))).
a(n) = 3*n*a(n-1) - 3*(n-1)^2*a(n-2) + (n-2)*(n-1)^2*a(n-3). (End)

A361617 a(n) = n! * Sum_{k=0..n} binomial(n+(n-1)*(k+1),n-k)/k!.

Original entry on oeis.org

1, 2, 15, 214, 4721, 146046, 5958367, 307382090, 19459587009, 1478414285146, 132440451881231, 13787717744245182, 1647673524863409265, 223671725058601427414, 34184743554559413628191, 5837132027535188545269106, 1106136052471647285563082497
Offset: 0

Views

Author

Seiichi Manyama, Mar 18 2023

Keywords

Links

Crossrefs

Main diagonal of A361616.
Cf. A361607.

Programs

  • PARI
    a(n) = n!*sum(k=0, n, binomial(n+(n-1)*(k+1), n-k)/k!);

Formula

a(n) = n! * [x^n] exp( x/(1-x)^n ) / (1-x)^n.
a(n) = Sum_{k=0..n} (n+(n-1)*(k+1))!/(n*k+n-1)! * binomial(n,k) for n > 0.
Showing 1-3 of 3 results.

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