A361572 -id:A361572 - cp's OEIS frontend

A293049 Expansion of e.g.f. exp(x^3/(1 - x)).

1, 0, 0, 6, 24, 120, 1080, 10080, 100800, 1149120, 14515200, 199584000, 2973801600, 47740492800, 820928908800, 15049152518400, 292919058432000, 6031865968128000, 130990787582054400, 2991455760887193600, 71659101232502784000, 1796424431562528768000

Offset: 0

Seiichi Manyama, Sep 29 2017

nonn

For n > 4, a(n) is a multiple of 10. - Muniru A Asiru, Oct 09 2017

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

Column k=2 of A293053.
E.g.f.: Product_{i>k} exp(x^i): A000262 (k=0), A052845 (k=1), this sequence (k=2), A293050 (k=3).
Cf. A361533, A361572.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*j!, j=3..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 30 2017
seq(factorial(k)*coeftayl(exp(x^3/(1-x)), x = 0, k),k=0..50); # Muniru A Asiru, Oct 09 2017

CoefficientList[Series[E^(x^3/(1-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2017 *)

x='x+O('x^66); Vec(serlaplace(exp(x^3/(1-x))))

E.g.f.: Product_{i>2} exp(x^i).
a(n) ~ n^(n-1/4) * exp(-5/2 + 2*sqrt(n) - n) / sqrt(2). - Vaclav Kotesovec, Sep 30 2017
a(n) = 2*(n-1) * a(n-1) - (n-1)*(n-2) * a(n-2) + 6*binomial(n-1,2) * a(n-3) - 12*binomial(n-1,3) * a(n-4) for n > 3. - Seiichi Manyama, Mar 15 2023
From Seiichi Manyama, Jun 17 2024: (Start)
a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k-1,n-3*k)/k!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=3..n} k * a(n-k)/(n-k)!. (End)

A361595 Expansion of e.g.f. exp( (x / (1-x))^3 ) / (1-x).

Original entry on oeis.org

1, 1, 2, 12, 120, 1320, 15480, 199080, 2862720, 46146240, 826156800, 16212873600, 344741443200, 7875365097600, 192137321376000, 4984375210214400, 136994756496998400, 3976455027389644800, 121533921410994892800, 3900447928934548992000

Offset: 0

Seiichi Manyama, Mar 16 2023

nonn

Cf. A002720, A361594.

Cf. A361572.

Table[n! * Sum[Binomial[n,3*k]/k!, {k,0,n/3}], {n,0,20}] (* Vaclav Kotesovec, Mar 17 2023 *)

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

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n,3*k)/k!.
From Vaclav Kotesovec, Mar 17 2023: (Start)
a(n) = (4*n - 3)*a(n-1) - 3*(n-1)*(2*n - 3)*a(n-2) + 2*(n-2)*(n-1)*(2*n - 3)*a(n-3) - (n-3)^2*(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 + (1511/2560)*3^(1/4)/n^(1/4)). (End)

A361576 Expansion of e.g.f. exp((x / (1-x))^4).

Original entry on oeis.org

1, 0, 0, 0, 24, 480, 7200, 100800, 1431360, 21772800, 370137600, 7185024000, 158150361600, 3848298854400, 100865282918400, 2799294930432000, 81599752346112000, 2492894621048832000, 79852538982408192000, 2684220785621286912000

Offset: 0

Seiichi Manyama, Mar 16 2023

nonn

Cf. A000262, A052887, A361572.

Table[n! * Sum[Binomial[n-1,n-4*k]/k!, {k,0,n/4}], {n,0,20}] (* Vaclav Kotesovec, Mar 17 2023 *)
With[{nn=20},CoefficientList[Series[Exp[(x/(1-x))^4],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Aug 01 2025 *)

my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((x/(1-x))^4)))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=4, i, (-1)^(j-4)*j*binomial(-4, j-4)*v[i-j+1]/(i-j)!)); v;

E.g.f.: exp( (x / (1-x))^4 ).
a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n-1,n-4*k)/k!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=4..n} (-1)^(k-4) * k * binomial(-4,k-4) * a(n-k)/(n-k)!.
a(n) = 5*(n-1)*a(n-1) - 10*(n-2)*(n-1)*a(n-2) + 10*(n-3)*(n-2)*(n-1)*a(n-3) - (n-3)*(n-2)*(n-1)*(5*n - 24)*a(n-4) + (n-5)*(n-4)*(n-3)*(n-2)*(n-1)*a(n-5). - Vaclav Kotesovec, Mar 17 2023

cp's OEIS Frontend

A293049 Expansion of e.g.f. exp(x^3/(1 - x)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A361595 Expansion of e.g.f. exp( (x / (1-x))^3 ) / (1-x).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A361576 Expansion of e.g.f. exp((x / (1-x))^4).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

PARI

Formula