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 A179201 E.g.f. equals the imaginary part of the i-th iteration of (x + x^2), where i=sqrt(-1). 2

%I

%S 0,0,2,-6,12,200,-6240,139440,-2869440,53386560,-708048000,

%T -6667689600,1162101600000,-68789252563200,3158414682259200,

%U -118988867559744000,3123174474201600000,17680394964750336000,-10490102782572441600000

%N E.g.f. equals the imaginary part of the i-th iteration of (x + x^2), where i=sqrt(-1).

%C Let H(x) equal the i-th iteration of (x + x^2), then

%C . the inverse of H(x) equals the conjugate of H(x);

%C . H(x+x^2) = H(x) + H(x)^2;

%C . H(x) = F(x) + i*G(x) where F(x) = e.g.f. of A179200 and G(x) = e.g.f. of this sequence, where H(F(x) - i*G(x)) = x;

%C . coefficients of H(x) form the first column of triangular matrix A030528 raised to the i-th power, where A030528(n,k) = C(k,n-k).

%F E.g.f.: G(x) satisfies:

%F . G(x) = sqrt( F(x) + F(x)^2 - F(x+x^2) )

%F . F(x) = (G(x+x^2)/G(x) - 1)/2

%F where F(x) is the e.g.f. of A179200.

%e E.g.f: G(x) = 2*x^2/2! - 6*x^3/3! + 12*x^4/4! + 200*x^5/5! +...

%e The e.g.f. of A179200, F(x), begins:

%e F(x) = x - 6*x^3/3! + 60*x^4/4! - 600*x^5/5! + 5880*x^6/6! - 38640*x^7/7! - 624960*x^8/8! +...

%e The i-th iteration of (x + x^2) = H(x) = F(x) + i*G(x), begins:

%e H(x) = x + i*x^2 - (1 + i)*x^3 + (5 + i)*x^4/2 - (15 - 5*i)*x^5/3 + (49 - 52*i)*x^6/6 - (23 - 83*i)*x^7/3 - (93 + 427*i)*x^8/6 + (15652 + 18537*i)*x^9/126 - (61567 + 24585*i)*x^10/126 + (369519 - 42094*i)*x^11/252 - (1743963 - 1222750*i)*x^12/504 + ...

%e where H(F(x) - i*G(x)) = x.

%o (PARI) {a(n)=local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial(c,r-c))),L=sum(k=1,#M,-(M^0-M)^k/k),N=sum(k=0,#L,(I*L)^k/k!));if(n<1,0,imag(n!*N[n,1]))}

%Y Cf. A179200, A030528.

%K sign

%O 0,3

%A _Paul D. Hanna_, Jul 02 2010

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Last modified July 24 06:56 EDT 2021. Contains 346273 sequences. (Running on oeis4.)