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.

A362188 Expansion of e.g.f. exp(x/(1-3*x)^(2/3)).

Original entry on oeis.org

1, 1, 5, 43, 513, 7781, 142861, 3075255, 75879553, 2110145833, 65275127541, 2222656864451, 82595058938305, 3325666654250253, 144214230714973693, 6700048267934377231, 331988586859256529921, 17475202293073669341905, 973765103770578798536293
Offset: 0

Views

Author

Seiichi Manyama, Apr 11 2023

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-3*x)^(2/3))))

Formula

a(n) = n! * Sum_{k=0..n} (-3)^k * binomial(-2*(n-k)/3,k)/(n-k)! = n! * Sum_{k=0..n} 3^(n-k) * binomial(n-k/3-1,n-k)/k!.

A362205 Expansion of e.g.f. exp(x/(1-3*x)^(1/3)).

Original entry on oeis.org

1, 1, 3, 19, 185, 2401, 38731, 745123, 16630769, 422157025, 12005107091, 377957000851, 13048046175913, 490052749100929, 19890724260375515, 867582126490694371, 40467070835396193761, 2009901604798183428673, 105901641663222888913699
Offset: 0

Views

Author

Seiichi Manyama, Apr 11 2023

Keywords

Crossrefs

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(x/(1-3*x)^(1/3))))

Formula

a(n) = n! * Sum_{k=0..n} (-3)^k * binomial(-(n-k)/3,k)/(n-k)! = n! * Sum_{k=0..n} 3^(n-k) * binomial(n-2*k/3-1,n-k)/k!.

A362166 Expansion of e.g.f. exp(-x * (1-3*x)^(1/3)).

Original entry on oeis.org

1, -1, 3, -1, 41, 299, 4531, 74507, 1474481, 33540119, 864507491, 24891022199, 791755864153, 27571976573699, 1043247441846611, 42615848603499779, 1869129393654945761, 87605345727468933167, 4369604246576366377411, 231091472431638655755119
Offset: 0

Views

Author

Seiichi Manyama, Apr 10 2023

Keywords

Crossrefs

Programs

  • Maple
    A362166 := proc(n)
        (-1)^n*n!*add(3^k * binomial((n-k)/3,k)/(n-k)!,k=0..n) ;
    end proc:
    seq(A362166(n),n=0..70) ; # R. J. Mathar, Dec 04 2023
  • PARI
    my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-x*(1-3*x)^(1/3))))

Formula

a(n) = (-1)^n * n! * Sum_{k=0..n} 3^k * binomial((n-k)/3,k)/(n-k)!.
D-finite with recurrence +(-9*n+11)*a(n) +2*(27*n^2-121*n+72)*a(n-1) +3*(-27*n^3+304*n^2-1053*n+1056)*a(n-2) +(-612*n^3+6984*n^2-23677*n+21227) *a(n-3) +4*(27*n-23)*(n-3)*a(n-4) -48*(9*n-10) *(n-3)*(n-4) *a(n-5) +64*(n-5)*(n-4)*(9*n^2-62*n+78)*a(n-6) +256*(n-5) *(n-6)*(17*n-24)*(n-4)*a(n-7)=0. - R. J. Mathar, Dec 04 2023
Showing 1-3 of 3 results.