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G.f. A(x) satisfies: A(x) = 1/(1-x)^2 + x^2*A'(x).
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%I #27 Oct 21 2023 06:14:05

%S 1,2,5,14,47,194,977,5870,41099,328802,2959229,29592302,325515335,

%T 3906184034,50780392457,710925494414,10663882416227,170622118659650,

%U 2900576017214069,52210368309853262,991996997887211999,19839939957744240002,416638739112629040065

%N G.f. A(x) satisfies: A(x) = 1/(1-x)^2 + x^2*A'(x).

%H Alois P. Heinz, <a href="/A143918/b143918.txt">Table of n, a(n) for n = 0..450</a>

%F a(n) = 3*floor(e*(n-1)!) - 1, n>1. - _Gary Detlefs_, Jun 10 2010

%F a(n) = (n-1) * a(n-1) + n + 1 for n > 0 and a(0) = 1. - _Werner Schulte_, Oct 20 2023

%F a(n) = A000522(n-1)*3 - 1, n > 0. - _M. F. Hasler_, Oct 20 2023

%e G.f.: A(x) = 1 + 2*x + 5*x^2 + 14*x^3 + 47*x^4 + 194*x^5 + 977*x^6 +...

%e x^2*A'(x) = 2*x^2 + 10*x^3 + 42*x^4 + 188*x^5 + 970*x^6 + 5862*x^7 +...

%p a:= proc(n) option remember; `if`(n<3, n^2+1,

%p ((n^2+1)*a(n-1)-(n-2)*(n+1)*a(n-2))/n)

%p end:

%p seq(a(n), n=0..25); # _Alois P. Heinz_, Jul 16 2017

%t a[0]=1; a[n_]:=(n-1)*a[n-1]+n+1; Array[a,23,0] (* _Stefano Spezia_, Oct 20 2023 *)

%o (PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=1/(1-x+x*O(x^n))^2+x^2*deriv(A)); polcoeff(A, n)}

%o (PARI) A143918_upto(N)=vector(N, n, N=if(n>1, (n-2)*N+n, 1)) \\ Gives the N initial values a(0..N-1). - _M. F. Hasler_, Oct 20 2023

%o (PARI) A143918(n)=if(n>1, localprec(max(logint(n=(n-1)!,10),5)+5); n\exp(-1)*3-1, n+1) \\ _M. F. Hasler_, Oct 20 2023

%Y Cf. A000522 (floor(e*n!) for n > 0).

%K nonn

%O 0,2

%A _Paul D. Hanna_, Sep 05 2008