A372199 - OEIS

%I #18 Apr 24 2024 06:07:21

%S 1,3,22,150,1370,14112,169884,2301264,34903584,584575200,10728401760,

%T 214047774720,4614042856320,106866549054720,2646889430976000,

%U 69814736722483200,1953778728154982400,57822137143219814400,1804373878844546150400,59213693468692224000000

%N a(n) = n! * F(n) * H(n), where F(n) is the n-th Fibonacci number and H(n) the n-th harmonic number.

%F E.g.f.: (5*x*log(-x^2 - x + 1) - sqrt(5)*(x - 2)*(log(2 - (sqrt(5) + 1)*x) -log((sqrt(5) - 1)*x + 2))) / (10*x*(x^2 + x - 1)).

%F a(n) = n! * A000045(n) * A001008(n) / A002805(n).

%F a(n) = A000045(n) * A000254(n) / A002805(n). - _R. J. Mathar_, Apr 24 2024

%F D-finite with recurrence 5*a(n) +5*(-2*n+1)*a(n-1) +(-5*n^2+10*n+1)*a(n-2) +(10*n^3-45*n^2+58*n-14)*a(n-3) +(5*n^4-40*n^3+109*n^2-108*n+16)*a(n-4) +2*(n-4)^3*a(n-5) +(n-4)^2*(n-5)^2*a(n-6)=0. - _R. J. Mathar_, Apr 24 2024

%p H := proc(n)

%p     add(1/i,i=1..n) ;

%p end proc:

%p A372199 := proc(n)

%p     n!*A000045(n)*H(n) ;

%p end proc:

%p seq(A372199(n),n=1..70) ; # _R. J. Mathar_, Apr 24 2024

%t a[n_] := n! Fibonacci[n] HarmonicNumber[n]; Array[a,20] (* _Stefano Spezia_, Apr 22 2024 *)

%Y Cf. A000045, A001008, A002805.

%K nonn

%O 1,2

%A _Vladimir Kruchinin_, Apr 21 2024