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.

A328055 Expansion of e.g.f. -log(1 - x / (1 - x)^2).

Original entry on oeis.org

0, 1, 5, 32, 270, 2904, 38400, 605520, 11113200, 232888320, 5488560000, 143704108800, 4138573824000, 130020673305600, 4425201196416000, 162194862064435200, 6369479157000960000, 266808274486161408000, 11874724379464826880000, 559591797303082672128000
Offset: 0

Views

Author

Ilya Gutkovskiy, Oct 03 2019

Keywords

Comments

a(n) is the number of ways to choose one element from each branch of labeled octopuses with n nodes (cf. A029767 and example below). - Enrique Navarrete, Oct 29 2023

Examples

			For n=2, the 3 labeled octopuses are the following, and there are 2+2+1 ways to choose one element from each branch:
O-1-2;
O-2-1;
1-O-2. - _Enrique Navarrete_, Oct 29 2023

Crossrefs

Cf. A001906, A004146, A005248, A005443, A029767, A052567 (exponential transform), A100404, A226968, A328054.

Programs

  • Magma
    [0] cat [Factorial(n - 1)*(Lucas(2*n)-2):n in [1..20]]; // Marius A. Burtea, Oct 03 2019
  • Mathematica
    nmax = 19; CoefficientList[Series[-Log[1 - x/(1 - x)^2], {x, 0, nmax}], x] Range[0, nmax]!
Join[{0}, Table[(n - 1)! (LucasL[2 n] - 2), {n, 1, 19}]]
  • PARI
    my(x='x+O('x^20)); concat(0, Vec(serlaplace(-log(1 - x / (1 - x)^2)))) \\ Michel Marcus, Oct 03 2019

Formula

E.g.f.: log(1 + Sum_{k>=1} Fibonacci(2*k) * x^k).
a(n) = (n - 1)! * (Lucas(2*n) - 2) for n > 0.

A366942 Expansion of e.g.f. 1/(1-x-2*x^2-3*x^3).

Original entry on oeis.org

1, 1, 6, 48, 408, 5040, 72000, 1184400, 22619520, 482993280, 11459750400, 299495750400, 8531976499200, 263353163673600, 8754879893760000, 311808414677760000, 11845876873678848000, 478163414336864256000, 20436460099541950464000, 921972301728418676736000
Offset: 0

Views

Author

Enrique Navarrete, Oct 29 2023

Keywords

Comments

a(n) is the number of ways to partition [n] into blocks of size at most 3, order the blocks, order the elements within each block, and choose 1 element from each block.
E.g.: a(4) = 408 since we have the following cases:
1,2,3,4: 24 such orderings, 1 way to choose one element from each block;
12,34: 24 such orderings, 2*2 ways to choose one element from each block;
12,3,4: 72 such orderings, 2*1*1 ways to choose one element from each block;
123,4: 48 such orderings, 3*1 ways to choose one element from each block;
so 24*1 + 24*4 + 72*2 + 48*3 = 408 ways.

Crossrefs

Cf. A000073, A000142, A052567, A052585, A101822, A189886, A364324.

Programs

  • Maple
    a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n, j)*j!*j, j=1..min(3, n)))
    end:
seq(a(n), n=0..19);  # Alois P. Heinz, Dec 14 2023
  • Mathematica
    With[{m = 20}, Range[0, m]! * CoefficientList[Series[1/(1 - x - 2*x^2 - 3*x^3), {x, 0, m}], x]] (* Amiram Eldar, Oct 30 2023 *)
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1/(1-x-2*x^2-3*x^3))) \\ Michel Marcus, Oct 30 2023

Formula

a(n) = A000142(n)*A101822(n).
a(n) = n*(a(n-1)+(n-1)*(2*a(n-2)+(n-2)*3*a(n-3))) for n>=3. - Alois P. Heinz, Dec 14 2023

A346432 a(0) = 1; a(n) = n! * Sum_{k=0..n-1} (n-k+1) * a(k) / k!.

Original entry on oeis.org

1, 2, 14, 144, 1968, 33600, 688320, 16450560, 449326080, 13806858240, 471395635200, 17703899136000, 725338710835200, 32193996432998400, 1538840509503897600, 78808952068374528000, 4305129487814098944000, 249876735246162984960000, 15356385691181506363392000
Offset: 0

Views

Author

Ilya Gutkovskiy, Jul 17 2021

Keywords

Crossrefs

Cf. A000670, A001339, A002866, A003480, A007840, A052555, A052567, A136658, A216794, A308939, A346433.

Programs

  • Mathematica
    a[0] = 1; a[n_] := a[n] = n! Sum[(n - k + 1) a[k]/k!, {k, 0, n - 1}]; Table[a[n], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/(2 - 1/(1 - x)^2), {x, 0, nmax}], x] Range[0, nmax]!
Table[Sum[(-1)^(n - k) StirlingS1[n, k] 2^k HurwitzLerchPhi[1/2, -k, 0]/2, {k, 0, n}], {n, 0, 18}]
  • PARI
    my(x='x+O('x^25)); Vec(serlaplace(1 / (2 - 1 / (1 - x)^2))) \\ Michel Marcus, Jul 18 2021

Formula

E.g.f.: 1 / (2 - 1 / (1 - x)^2).
E.g.f.: 1 / (1 - Sum_{k>=1} (k+1) * x^k).
a(0) = 1, a(1) = 2, a(2) = 14; a(n) = 4 * n * a(n-1) - 2 * n * (n-1) * a(n-2).
a(n) = Sum_{k=0..n} (-1)^(n-k) * Stirling1(n,k) * 2^k * A000670(k).
a(n) = n! * A003480(n).
Showing 1-3 of 3 results.

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