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%I #10 Mar 09 2023 09:00:11
%S 0,1,-2,12,-96,880,-8720,90752,-975936,10737152,-120093056,1360051456,
%T -15556087296,179424700416,-2084953411584,24393551634432,
%U -287204585508864,3400978267127808,-40480500900446208,484006813958356992,-5810240353159839744,70001749695581061120
%N G.f. A(x) satisfies x = A( A(x) + 4*A(x)^2 ).
%C Composition inverse of A027436.
%H G. C. Greubel, <a href="/A307103/b307103.txt">Table of n, a(n) for n = 0..485</a>
%F a(n) = (-1)^(n+1) * A213422(n).
%e G.f. = x - 2*x^2 + 12*x^3 - 96*x^4 + 880*x^5 - 8720*x^6 + 90752*x^7 + ...
%t a[ n_] := Module[ {A, x}, A = x; Do[ A += x O[x]^k; A = Normal[A] + x^k ((-4)^(k-1) CatalanNumber[k-1] - SeriesCoefficient[ ComposeSeries[A, A], k])/2, {k, 2, n}]; oefficient[A, x, n]];
%t (* Second program *)
%t T[n_, k_]:= T[n, k]= If[k<0 || k>n, 0, If[n==0, 0, If[k==n, 1, 2^(2*n - 2*k-1)*(k/n)*Binomial[2*n-k-1, n-1] - (1/2)*Sum[T[n, n-j-1]*T[n-j-1, k], {j,0,n-k-2}] ]]];
%t a[n_]:= (-1)^(n+1)*T[n,1];
%t Table[a[n], {n,0,30}] (* _G. C. Greubel_, Mar 08 2023 *)
%o (PARI) {a(n) = my(A); A = x; for(k=2, n, A += x*O(x^k); A = truncate(A) + x^k * ((-4)^(k-1) * binomial(2*k-2,k-1)/k - polcoeff(subst(A, x, A), k))/2); polcoeff(A, n)};
%o (SageMath)
%o @CachedFunction
%o def T(n,k):
%o if (k<0 or k>n): return 0
%o elif (n==0): return 0
%o elif (k==n): return 1
%o else: return 2^(2*n-2*k-1)*(k/(2*n-k))*binomial(2*n-k, n) - (1/2)*sum( T(n, n-j-1)*T(n-j-1, k) for j in range(n-k-1) )
%o def A307103(n): return (-1)^(n+1)*T(n,1)
%o [A307103(n) for n in range(31)] # _G. C. Greubel_, Mar 08 2023
%Y Cf. A027436, A213422.
%K sign
%O 0,3
%A _Michael Somos_, Mar 24 2019
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