A361576 -id:A361576 - cp's OEIS frontend

A293050 Expansion of e.g.f. exp(x^4/(1 - x)).

1, 0, 0, 0, 24, 120, 720, 5040, 60480, 725760, 9072000, 119750400, 1756339200, 28021593600, 479480601600, 8717829120000, 168254102016000, 3438311804928000, 74160828758016000, 1682757222322176000, 40061786401308672000, 998402161605488640000

Offset: 0

Seiichi Manyama, Sep 29 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..444

Column k=3 of A293053.
E.g.f.: Product_{i>k} exp(x^i): A000262 (k=0), A052845 (k=1), A293049 (k=2), this sequence (k=3).
Cf. A361545, A361576.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*j!, j=4..n))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 29 2017

a[n_] := a[n] = If[n==0, 1, Sum[a[n-j] Binomial[n-1, j-1] j!, {j, 4, n}]];
a /@ Range[0, 23] (* Jean-François Alcover, Dec 21 2020, after Alois P. Heinz *)

x='x+O('x^66); Vec(serlaplace(exp(x^4/(1-x))))

E.g.f.: Product_{i>3} exp(x^i).
From Vaclav Kotesovec, Sep 30 2017: (Start)
a(n) = 2*(n-1)*a(n-1) - (n-2)*(n-1)*a(n-2) + 4*(n-3)*(n-2)*(n-1)*a(n-4) - 3*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5).
a(n) ~ n^(n-1/4) * exp(-7/2 + 2*sqrt(n) - n) / sqrt(2).
(End)
From Seiichi Manyama, Jun 17 2024: (Start)
a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n-3*k-1,n-4*k)/k!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=4..n} k * a(n-k)/(n-k)!. (End)

A361572 Expansion of e.g.f. exp( (x / (1-x))^3 ).

Original entry on oeis.org

1, 0, 0, 6, 72, 720, 7560, 90720, 1270080, 20381760, 364694400, 7125148800, 150186960000, 3393726336000, 81882210009600, 2102315389574400, 57244753133568000, 1647544166940672000, 49957730917981286400, 1591303422125646028800

Offset: 0

Seiichi Manyama, Mar 16 2023

nonn

Cf. A000262, A052887, A361576.

Cf. A091695.

Table[n! * Sum[Binomial[n-1,n-3*k]/k!, {k,0,n/3}], {n,0,20}] (* Vaclav Kotesovec, Mar 17 2023 *)
With[{nn=20},CoefficientList[Series[Exp[(x/(1-x))^3],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Sep 07 2024 *)

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

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=3, i, (-1)^(j-3)*j*binomial(-3, j-3)*v[i-j+1]/(i-j)!)); v;

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-1,n-3*k)/k!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=3..n} (-1)^(k-3) * k * binomial(-3,k-3) * a(n-k)/(n-k)!.
From Vaclav Kotesovec, Mar 17 2023: (Start)
a(n) = 4*(n-1)*a(n-1) - 6*(n-2)*(n-1)*a(n-2) + (n-2)*(n-1)*(4*n - 9)*a(n-3) - (n-4)*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 3^(1/8) * exp(-1/4 + 5*3^(-1/4)*n^(1/4)/8 - sqrt(3*n)/2 + 4*3^(-3/4)*n^(3/4) - n) * n^(n - 1/8) / 2 * (1 - (409/2560)*3^(1/4)/n^(1/4)). (End)

cp's OEIS Frontend

A293050 Expansion of e.g.f. exp(x^4/(1 - x)).

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A361572 Expansion of e.g.f. exp( (x / (1-x))^3 ).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula