A289085 - OEIS

%I #18 Aug 18 2023 23:36:21

%S 0,1,2,4,8,12,-36,-656,-6016,-45712,-303584,-1614784,-3151424,

%T 89449152,2020752864,30106674944,371094759424,3803441275648,

%U 27086999353856,-53440551394304,-7360216885479424,-195653223115035648,-3852848368364645376,-62648371228429684736

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

%C Here, i is the imaginary unit. The complex integer sequence c(n) = A289084(n) + i*A289085(n) is one of a family of sequences whose e.g.f.s satisfy the differential equation f''(z) = f'(z)f(z). For more details, see A289064 and A289082.

%H Stanislav Sykora, <a href="/A289085/b289085.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.: imag(2*L0*tan(L0*z + L1)), where L0 = sqrt(i/2-1) and L1 = arccos(sqrt(1+2*i)).

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

%o (PARI) c0=2; 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   imag(a)

%Y Cf. A289084 (real 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), A289082 and A289083 (1,i), A289086 and A289087 (1,2i), A289088 and A289089 (2,2i).

%K sign

%O 0,3

%A _Stanislav Sykora_, Jul 19 2017