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A226952 Triangle of coefficients of Faber polynomials for (3*x - sqrt(x^2 - 4*x))/2.

%I A226952 #15 Sep 08 2022 08:46:05
%S A226952 0,-1,1,-1,-2,1,-4,0,-3,1,-13,-4,2,-4,1,-46,-10,-5,5,-5,1,-166,-36,-6,
%T A226952 -8,9,-6,1,-610,-126,-28,0,-14,14,-7,1,-2269,-456,-92,-24,10,-24,20,
%U A226952 -8,1,-8518,-1674,-333,-63,-27,27,-39,27,-9,1
%N A226952 Triangle of coefficients of Faber polynomials for (3*x - sqrt(x^2 - 4*x))/2.
%H A226952 G. C. Greubel, <a href="/A226952/b226952.txt">Rows n = 0..100 of triangle, flattened</a>
%F A226952 G.f.: log(1 + (1 - sqrt(1-4*t))/2 - t*x) = Sum_{n>0} Sum_{k=0..n} T(n,k) * x^k * t^n/n.
%F A226952 T(n,k) = n*Sum_{j=1..n-k} binomial(j+k,k)*(j)*binomial(2*(n-k)-j-1, n-k-1)*(-1)^j/((j+k)*(n-k)), k<n, T(0,0)=0, T(n,n)=1.
%F A226952 (-1)^(n+1) * Sum_{k=0..n} T(n,k) = 2*A181933(n).
%F A226952 T(n,0) = -A026641(n-1), n>0.
%e A226952 Triangle begins as:
%e A226952     0;
%e A226952    -1,   1;
%e A226952    -1,  -2,  1;
%e A226952    -4,   0, -3,  1;
%e A226952   -13,  -4,  2, -4,  1;
%e A226952   -46, -10, -5,  5, -5,  1;
%t A226952 T[n_,k_]:= If[n==k==0, 0, If[k==n, 1, n*Sum[(-1)^j*j*Binomial[j+k, k]* Binomial[2*n-2*k-j-1, n-k-1]/((j+k)*(n-k)), {j, 1, n-k}]]]; Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* _G. C. Greubel_, Apr 29 2019 *)
%o A226952 (Maxima) T(n,k):=if n=0 and k=0 then 0 else if n=k then 1 else n*sum(binomial(i+k,k)*(i)*binomial(2*(n-k)-i-1,n-k-1)*(-1)^(i)/((i+k)*(n-k)),i,1,n-k);
%o A226952 (PARI) {T(n,k) = if(n==0 && k==0, 0, if(k==n, 1, n*sum(j=1,n-k, (-1)^j*j* binomial(j+k, k)*binomial(2*n-2*k-j-1, n-k-1)/((j+k)*(n-k)))))}; \\ _G. C. Greubel_, Apr 29 2019
%o A226952 (Magma) [[n eq 0 and k eq 0 select 0 else k eq n select 1 else n*(&+[ (-1)^j*j*Binomial(j+k,k)*Binomial(2*n-2*k-j-1, n-k-1)/((j+k)*(n-k)): j in [1..n-k]]): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Apr 29 2019
%o A226952 (Sage)
%o A226952 def T(n, k):
%o A226952   if (k==n==0): return 0
%o A226952   elif (k==n): return 1
%o A226952   else: return n*sum((-1)^j*j* binomial(j+k, k)*binomial(2*n-2*k-j-1, n-k-1)/((j+k)*(n-k)) for j in (1..n-k))
%o A226952 [[T(n, k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Apr 29 2019
%K A226952 sign,tabl
%O A226952 0,5
%A A226952 _Dmitry Kruchinin_, Jun 24 2013