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A213190 a(0)=1, a(1)=1, a(n) = n*a(n-1) + 3*a(n-2).

%I A213190 #52 Feb 19 2025 01:03:46
%S A213190 1,1,5,18,87,489,3195,23832,200241,1873665,19337373,218332098,
%T A213190 2677997295,35468961129,504599447691,7675398598752,124320175923105,
%U A213190 2136469186489041,38829405884572053,744168119366336130,14999850604980438759,317229367062688222329
%N A213190 a(0)=1, a(1)=1, a(n) = n*a(n-1) + 3*a(n-2).
%C A213190 Let S(i,j,x,n) denote a sequence of the form a(0) = i, a(1) = j, a(n) = n*a(n-1) + x*a(n-2). Then S(i,j,x,n) = i*(Sum_{k=0..floor(n/2+1)} (n-k-2)!*binomial(n-k,k+2)*x^(k+1)/k!) + j*(Sum_{k=0..floor((n+1)/2)} (n-k)!*binomial(n-k-1,k)*x^k/(k+1)!), n>1.
%C A213190 Other sequences of this form are:
%C A213190   S(0,1,1,n): A001053(n+1),
%C A213190   S(1,0,1,n): A058307(n+1),
%C A213190   S(1,1,1,n): A001040.
%C A213190 For a more advanced treatment of this topic see A084950 and A221913.
%C A213190 In cases where i = j=1 the above formula can be reduced to S(1,1,x,n) = Sum_{k =0..floor(n/2)} (((n-k)!/k!)*binomial(n-k,k)*x^k). - _Gary Detlefs_ and _Wolfdieter Lang_, Mar 06 2013
%C A213190 From _Wolfdieter Lang_, Mar 08 2013: (Start)
%C A213190 a(n) = Q(n,3), the row polynomial of A084950 evaluated for x=3, is the denominator of the n-th approximation to the continued fraction 1/(1+3/(2+3/(3+3/(4+... The corresponding numerator is A222472(n). The limit of this continued fraction is (z/6)*BesselI(1,z)/BesselI(0,z), with z= 2*sqrt(3) which is 0.4845161749874040...
%C A213190 E.g.f.: a(n) = (Pi*x/(2*sqrt(1-z)))*(BesselJ(1, x*sqrt(1-z))*BesselY(0,x) - BesselY(1, x*sqrt(1-z))*BessJ(0,x)) with x := 2*sqrt(-3).
%C A213190 For a combinatorial interpretation in terms of labeled Morse codes see a comment and reference on A084950. Here each dash has label x = 3, and the dots have label j if they are at position j. Labels are multiplied and for a(n) all labeled codes on [1,2,...,n] have to be summed. (End)
%H A213190 G. C. Greubel, <a href="/A213190/b213190.txt">Table of n, a(n) for n = 0..445</a> (terms 0..250 from Reinhard Zumkeller)
%F A213190 a(0)=1, a(1)=1, a(n) = (Sum_{k=0..floor(n/2+1)} (n-k-2)!*binomial(n-k,k+2)*3^(k+1)/k!) + (Sum_{k=0..floor((n+1)/2)} (n-k)!*binomial(n-k-1,k)*3^k/(k+1)!), n>1.
%F A213190 a(n) = 2*sqrt(3)^(n+1)*(BesselI(0, 2*sqrt(3))*BesselK(n+1, 2*sqrt(3)) - BesselK(0, 2*sqrt(3))*BesselI(n+1, -2*sqrt(3))), n >= 0. See Q(n,3) from A084950. - _Wolfdieter Lang_, Mar 06 2013
%F A213190 a(n) = Sum_{k=0..floor(n/2)} (((n-k)!/k!)*binomial(n-k,k)*3^k). - _Gary Detlefs_ and _Wolfdieter Lang_, Mar 06 2013
%F A213190 Asymptotics: lim_{n->oo} a(n)/n! = BesselI(0,2*sqrt(3)) = 7.15899653680... See a comment on the large n behavior of Bessel functions under A084950. - _Wolfdieter Lang_, Mar 08 2013
%e A213190 From _Wolfdieter Lang_, Mar 08 2013: (Start)
%e A213190 a(4) = 4*a(3) + 3*a(2) = 4*18 + 3*5 = 87.
%e A213190 Morse code: a(4) = 87 from the sum of all 5 labeled codes on [1,2,3,4], one with no dash, three with one dash and one with two dashes: 4! + (3*4 + 1*4 + 1*2)*(3) + (3)^2 = 87. (End)
%p A213190 A:=(n,x)->sum((n-k-2)!*binomial(n-k,k+2)*x^(k+1)/k!,k=0..floor(n/2+1))
%p A213190 B:=(n,x)->sum(n-k)!*binomial(n-k-1,k)*x^k/(k+1)!,k=0..floor((n+1)/2))
%p A213190 seq(A(n,3)+B(n,3), n=2..20)
%t A213190 RecurrenceTable[{a[0] == 1, a[1] == 1, a[n] == n*a[n - 1] + 3 a[n - 2]}, a[n], {n, 50}] (* _G. C. Greubel_, Aug 16 2017 *)
%o A213190 (PARI) a(n) = sum(k=0, n\2, ((n-k)!/k!)*binomial(n-k,k)*3^k); /* _Joerg Arndt_, Mar 07 2013 */
%o A213190 (Haskell)
%o A213190 a213190 n = a213190_list !! n
%o A213190 a213190_list = 1 : 1 : zipWith (+)
%o A213190    (zipWith (*) [2..] $ tail a213190_list) (map (* 3) a213190_list)
%o A213190 -- _Reinhard Zumkeller_, Feb 20 2015
%Y A213190 Cf. A001053, A058307, A001040, A084950, A221913.
%K A213190 nonn
%O A213190 0,3
%A A213190 _Gary Detlefs_, Feb 28 2013