A193930 E.g.f.: exp(x+x^2+x^3+x^4).

1, 1, 3, 13, 73, 381, 2611, 19993, 165873, 1436473, 14004451, 145099461, 1584090553, 18196817653, 223416271443, 2865429498961, 38330181602401, 535448870264433, 7823019065848003, 118402856414023933, 1856454825152993961, 30160691907215561581

Offset: 0

Author

Vladimir Kruchinin, Aug 09 2011

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..524

Crossrefs

Cf. A047974, A118589.

Programs

Magma

I:=[1,1,3,13]; [n le 4 select I[n] else Self(n-1)+2*(n-2)*Self(n-2)+3*(n-2)*(n-3)*Self(n-3)+4*(n-2)*(n-3)*(n-4)*Self(n-4): n in [1..25]]; // Vincenzo Librandi, Oct 12 2014

Maple

A193930:=n->`if`(n=0,1,n!*add(add( (-1)^i*binomial(k,k-i)*binomial(n-4*i-1,k-1)/k!, i=0..(n-k)/4), k=1..n)): seq(A193930(n), n=0..20); # Wesley Ivan Hurt, Oct 11 2014
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*j!, j=1..min(n, 4)))
    end:
seq(a(n), n=0..23);  # Alois P. Heinz, Sep 29 2017

Mathematica

RecurrenceTable[{a[n] == a[n - 1] + 2 (n - 1) a[n - 2] + (n - 1) (n - 2) 3 a[n - 3] + (n - 1) (n - 2) (n - 3) 4 a[n - 4], a[0] == 1,
a[1] == 1, a[2] == 3, a[3] == 13}, a, {n, 0, 20}] (* Geoffrey Critzer, Oct 11 2014 *)
With[{nn=30},CoefficientList[Series[Exp[x+x^2+x^3+x^4],{x,0,nn}],x] Range[ 0,nn]!] (* Harvey P. Dale, Aug 20 2021 *)

Maxima

a(n):=if n=0 then 1 else n!*sum(sum((-1)^i*binomial(k,k-i)*binomial(n-4*i-1,k-1),i,0,(n-k)/4)/k!,k,1,n);

PARI

x = y + O(y^50); concat(0, Vec(serlaplace(exp(x+x^2+x^3+x^4)))) \\ Michel Marcus, Oct 12 2014

Formula

a(n) = n!*sum(k=1..n, sum(i=0..(n-k)/4, (-1)^i*binomial(k,k-i)*binomial(n-4*i-1,k-1))/k!), n>0, a(0)=1.

a(n) = a(n-1) + 2*(n-1)*a(n-2) + 3*(n-1)(n-2)*a(n-3) + 4*(n-1)*(n-2)*(n-3)*a(n-4). - Geoffrey Critzer, Oct 11 2014