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.

A356815 Expansion of e.g.f. exp(-x * (exp(2*x) + 1)).

Original entry on oeis.org

1, -2, 0, 4, 32, 48, -608, -6400, -24064, 163072, 3567104, 28394496, 6535168, -3250745344, -50725740544, -344530853888, 2476610551808, 110057610608640, 1655672654135296, 9616664975114240, -195178079811272704, -6998474114188967936, -110894925369151848448
Offset: 0

Views

Author

Seiichi Manyama, Aug 29 2022

Keywords

Links

Crossrefs

Cf. A356816, A356818.
Cf. A240165, A351736, A356812.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*(exp(2*x)+1))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k/(1-(2*k-1)*x)^(k+1)))
  • PARI
    a(n) = sum(k=0, n, (-1)^k*(2*k-1)^(n-k)*binomial(n, k));

Formula

G.f.: Sum_{k>=0} (-x)^k / (1 - (2*k-1)*x)^(k+1).
a(n) = Sum_{k=0..n} (-1)^k * (2*k-1)^(n-k) * binomial(n,k).

A356816 Expansion of e.g.f. exp(-x * (exp(3*x) + 1)).

Original entry on oeis.org

1, -2, -2, 1, 88, 583, 676, -35597, -519392, -3359393, 19013884, 896435395, 13640180896, 85591357135, -1527872118356, -61100053650053, -1076294742932288, -7610985095240513, 200631806070276988, 9284475508083767059, 200226297062313730816, 1940767272243466116463
Offset: 0

Views

Author

Seiichi Manyama, Aug 29 2022

Keywords

Links

Crossrefs

Cf. A356815, A356818.
Cf. A351737, A356813.

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-x*(exp(3*x)+1))))
  • PARI
    my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (-x)^k/(1-(3*k-1)*x)^(k+1)))
  • PARI
    a(n) = sum(k=0, n, (-1)^k*(3*k-1)^(n-k)*binomial(n, k));

Formula

G.f.: Sum_{k>=0} (-x)^k / (1 - (3*k-1)*x)^(k+1).
a(n) = Sum_{k=0..n} (-1)^k * (3*k-1)^(n-k) * binomial(n,k).

A356817 a(n) = Sum_{k=0..n} (-1)^k * (k*n-1)^(n-k) * binomial(n,k).

Original entry on oeis.org

1, -2, 0, 1, 144, 4143, 110368, 2535475, 13299968, -5169863825, -639341093376, -59073970497885, -4677854594527232, -276406098219258425, 2399871442122924032, 5163244810691492730907, 1331213942683118587674624, 262517264591996332314037215
Offset: 0

Views

Author

Seiichi Manyama, Aug 29 2022

Keywords

Crossrefs

Cf. A356815, A356816, A356818.
Cf. A356806, A356811, A356814.

Programs

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

Formula

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

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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