A371114 - OEIS

%I #26 Mar 29 2024 05:48:17

%S 0,1,1,-1,-16,-111,-691,-4145,-23080,-96159,195137,13914911,284958904,

%T 4842967921,77613841629,1219132694767,19006810258064,294117291312577,

%U 4469910552829473,64942899785556031,841752172982238304,7465153745073705041,-61832090598783228403,-6408471053640082778097

%N a(n) is n! times the coefficient of x in the associated Laguerre polynomial Laguerre(n,x,1).

%F a(n) = n!*Sum_{k=0..n-1} A009940(k)/(k!*(n-k)).

%F a(n) = (4*n - 7)*a(n-1) - (6*n^2 - 25*n + 27)*a(n-2) + (n-2)*(4*n^2 - 21*n + 28)*a(n-3) - (n-3)^3*(n-2)*a(n-4). - _Vaclav Kotesovec_, Mar 12 2024

%t a[n_]:=n! D[LaguerreL[n,x,1],x]/.{x->0}; Array[a,25,0]

%t Table[n! Sum[LaguerreL[k,1]/(n-k),{k,0,n-1}],{n,0,25}]

%t RecurrenceTable[{(-3 + n)^3*(-2 + n)*a[n-4] - (-2 + n)*(28 - 21*n + 4*n^2)*a[n-3] + (27 - 25*n + 6*n^2)*a[n-2] + (7 - 4*n)*a[n-1] + a[n] == 0, a[0] == 0, a[1] == 1, a[2] == 1, a[3] == -1}, a, {n, 0, 20}] (* _Vaclav Kotesovec_, Mar 12 2024 *)

%o (PARI) a(n) = n!*sum(k=0, n-1, pollaguerre(k, 0, 1)/(n-k)); \\ _Michel Marcus_, Mar 12 2024

%Y Cf. A009940.

%K sign

%O 0,5

%A _Rui Xian Siew_, Mar 10 2024