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.

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A362300 a(n) = n! * Sum_{k=0..floor(n/3)} (n/3)^k * binomial(n-2*k,k)/(n-2*k)!.

Original entry on oeis.org

1, 1, 1, 7, 33, 101, 1681, 14211, 72577, 1906633, 23242401, 166218911, 5966236321, 95016917997, 873707885233, 39767572858651, 781865428682241, 8787169718273681, 484500265577706817, 11335266937098816183, 150554918241183405601, 9749671976020428623221
Offset: 0

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Author

Seiichi Manyama, Apr 15 2023

Keywords

Comments

Let k be a positive integer. It appears that reducing this sequence modulo k produces an eventually periodic sequence with period a multiple of k. For example, modulo 9 the sequence becomes [1, 1, 1, 7, 6, 2, 7, 0, 1, 1, 0, 8, 7, 0, 7, 7, 0, 8, 1, 0, 1, 7, 0, 2, 7, 0, 1, 1, 0, 8, 7, 0, 7, 7, 0, 8, 1, 0, 1, 7, 0, 2, 7, 0, 1, ...], with an apparent period [2, 7, 0, 1, 1, 0, 8, 7, 0, 7, 7, 0, 8, 1, 0, 1, 7, 0] of length 18 starting at a(5). - Peter Bala, Apr 16 2023

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((-lambertw(-x^3))^(1/3))/(1+lambertw(-x^3))))

Formula

a(n) = A362043(n,2*n).
a(n) = n! * [x^n] exp(x + n*x^3/3).
E.g.f.: exp( ( -LambertW(-x^3) )^(1/3) ) / (1 + LambertW(-x^3)).
a(n) ~ (1 + 2*cos(2*Pi*mod(n,3)/3 - sqrt(3)/2)/exp(3/2)) * n^n / (sqrt(3) * exp(2*n/3 - 1)). - Vaclav Kotesovec, Apr 18 2023

A362283 Expansion of e.g.f. exp( sqrt(-LambertW(-x^2)) ).

Original entry on oeis.org

1, 1, 1, 4, 13, 106, 601, 7456, 60649, 1012348, 10748161, 225641296, 2957978101, 74847384184, 1168123938073, 34598428916416, 626497273410961, 21261683280971536, 438222313050326209, 16765636110497697088, 387549609831150094621, 16502188154766906299296
Offset: 0

Views

Author

Seiichi Manyama, Apr 14 2023

Keywords

Crossrefs

Programs

  • PARI
    my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(sqrt(-lambertw(-x^2)))))

Formula

a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/2)} A034940(k) * binomial(n-1,2*k) * a(n-2*k-1).
Showing 1-2 of 2 results.