A190875 - OEIS

%I #17 Sep 06 2025 01:39:02

%S 1,1,1,1,25,121,361,841,21841,184465,912241,3334321,89824681,

%T 1063799881,7325422105,36454050361,1017339566881,15314809967521,

%U 134865698721121,848781820990945,24508030788375481,443521231499750041,4748434334677932361,36119410438503259561

%N E.g.f. exp(x+x^4).

%H Vincenzo Librandi, <a href="/A190875/b190875.txt">Table of n, a(n) for n = 0..105</a>

%F a(n) = n!*Sum_{j=0..n/3} binomial(n-3*j,j)/(n-3*j)!.

%F Recurrence: a(n) = a(n-1) + 4*(n-3)*(n-2)*(n-1)*a(n-4). - _Vaclav Kotesovec_, Jun 26 2013

%F a(n) ~ 2^(n/2-1)*exp(n^(1/4)/sqrt(2)-3*n/4)*n^(3*n/4). - _Vaclav Kotesovec_, Jun 26 2013

%t CoefficientList[Series[E^(x+x^4), {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Jun 26 2013 *)

%o (Maxima) a(n):=n!*sum(binomial(n-3*j,j)/(n-3*j)!,j,0,n/3);

%o (PARI) Vec(serlaplace(exp(x+x^4+O(x^66)))) /* _Joerg Arndt_, May 24 2011 */

%K nonn

%O 0,5

%A _Vladimir Kruchinin_, May 22 2011