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A289082 Real parts of the recursive sequence a(n+2) = Sum_{k=0..n} binomial(n,k)*a(k)*a(n+1-k), with a(0)=1, a(1)=i. 8

%I #24 Aug 18 2023 23:36:00

%S 1,0,0,-1,-4,-11,-26,-23,376,4041,28266,153973,517204,-1969651,

%T -62317666,-796305583,-7467806224,-49765963119,-74804319534,

%U 4766716058093,107039955050444,1549401029915989,16531537283552134,99639596341492297,-903257832030151064,-44705398895606766759

%N Real parts of the recursive sequence a(n+2) = Sum_{k=0..n} binomial(n,k)*a(k)*a(n+1-k), with a(0)=1, a(1)=i.

%C Here, i is the imaginary unit. The complex integer sequence c(n) = A289082(n) + i*A289083(n) is one of a family of sequences whose e.g.f.s satisfy the differential equation f''(z) = f'(z)f(z). Each such sequence is uniquely characterized by its two starting terms, which may also be complex integers. For more details, see A289064.

%H Stanislav Sykora, <a href="/A289082/b289082.txt">Table of n, a(n) for n = 0..200</a>

%H S. Sykora, <a href="http://dx.doi.org/10.3247/SL6Math17.001">Sequences related to the differential equation f'' = af'f</a>, Stan's Library, Vol. VI, Jun 2017.

%F E.g.f.: real(2*L0*tan(L0*z + L1)), where L0 = sqrt(i/2-1/4) and L1 = arccos(sqrt(i/2+1)).

%t a[0]=1; a[1]=I; a[n_]:=a[n]=Sum[Binomial[n - 2, k] a[k] a[n - 1 - k], {k, 0, n - 2}]; Re[Table[a[n], {n, 0, 50}]] (* _Indranil Ghosh_, Jul 20 2017 *)

%o (PARI) c0=1; c1=I; nmax = 200;

%o a=vector(nmax+1); a[1]=c0; a[2]=c1;

%o for(m=0, #a-3, a[m+3]=sum(k=0, m, binomial(m, k)*a[k+1]*a[m+2-k]));

%o real(a)

%Y Cf. A289083 (imaginary part).

%Y Sequences for other starting pairs: A000111 (1,1), A289064 (1,-1), A289065 (2,-1), A289066 (3,1), A289067 (3,-1), A289068 (1,-2), A289069 (3,-2), A289070 (0,3), A289084 and A289085 (2,i), A289086 and A289087 (1,2i), A289088 and A289089 (2,2i).

%K sign

%O 0,5

%A _Stanislav Sykora_, Jul 19 2017

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)