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%I A097090 #25 Oct 29 2015 10:10:06
%S A097090 1,2,-2,6,-22,80,-228,18,6694,-65604,396804,-1336332,-2510244,
%T A097090 64799884,-302351144,-1410221598,23754923526,16833211660,
%U A097090 -2598949277964,14767224078420,229529725999500,-3478598282993328,-13287810766972728,667271251276705140,-1630867775606147844
%N A097090 G.f. A(x) satisfies: A(A(x)) = x*(1+2*x)^2.
%H A097090 Paul D. Hanna, <a href="/A097090/b097090.txt">Table of n, a(n) for n = 1..130</a>
%H A097090 Dmitry Kruchinin, Vladimir Kruchinin, <a href="http://arxiv.org/abs/1302.1986">Method for solving an iterative functional equation $A^{2^n}(x)=F(x)$</a>, arXiv:1302.1986
%F A097090 a(n)=T(n,1), T(n,m)=if n=m then 1 else 1/2*(binomial(2*m,n-m)*2^(n-m)-sum(i=m+1..n-1, T(n,i)*T(i,m))). [From Vladimir Kruchinin, Nov 10 2011]
%e A097090 G.f.: A(x) = x + 2*x^2 - 2*x^3 + 6*x^4 - 22*x^5 + 80*x^6 - 228*x^7 + 18*x^8 +...
%e A097090 where A(A(x)) = x + 4*x^2 + 4*x^3.
%e A097090 Illustrate Vladimir Kruchinin's formula by the triangular matrix T:
%e A097090 1;
%e A097090 2, 1;
%e A097090 -2, 4, 1;
%e A097090 6, 0, 6, 1;
%e A097090 -22, 4, 6, 8, 1;
%e A097090 80, -16, 2, 16, 10, 1;
%e A097090 -228, 48, -6, 8, 30, 12, 1;
%e A097090 18, -12, 0, 0, 30, 48, 14, 1;
%e A097090 6694, -1460, 232, -32, 10, 76, 70, 16, 1; ...
%e A097090 in which the g.f. of column k = A(x)^k and A(x) is the g.f. of this sequence.
%e A097090 The matrix square, T^2, of the above triangle begins:
%e A097090 1;
%e A097090 4, 1;
%e A097090 4, 8, 1;
%e A097090 0, 24, 12, 1;
%e A097090 0, 32, 60, 16, 1;
%e A097090 0, 16, 160, 112, 20, 1;
%e A097090 0, 0, 240, 448, 180, 24, 1;
%e A097090 0, 0, 192, 1120, 960, 264, 28, 1;
%e A097090 0, 0, 64, 1792, 3360, 1760, 364, 32, 1; ...
%e A097090 in which the g.f. column k = (x + 4*x^2 + 4*x^3)^k = sum(n>=k, binomial(2*k,n-k)*2^(n-k)*x^n).
%t A097090 T[n_, m_] := T[n, m] = If[n == m, 1, (1/2)*(Binomial[2 m, n - m]*2^(n - m) - Sum[T[n, i]*T[i, m], {i, m+1, n-1}])]; a[n_] := T[n, 1] // Numerator; Table[a[n], {n, 1, 25}] (* _Jean-François Alcover_, Oct 29 2015, after _Vladimir Kruchinin_ *)
%o A097090 (PARI) {a(n)=local(A,B,F);F=x*(1+2*x+x*O(x^n))^2;A=F; for(i=0,n,B=serreverse(A);A=(A+subst(B,x,F))/2);polcoeff(A,n,x)}
%o A097090 (PARI) /* Vladimir Kruchinin's formula: a(n) = T(n,1) where: */
%o A097090 {T(n,k)=if(n==k,1,1/2*(binomial(2*k,n-k)*2^(n-k)-sum(j=k+1,n-1,T(n,j)*T(j,k))))}
%o A097090 {a(n)=T(n,1)} /* Paul D. Hanna, Nov 10 2011 */
%Y A097090 Cf. A107099.
%K A097090 sign
%O A097090 1,2
%A A097090 _Paul D. Hanna_, Jul 23 2004

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