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

0, -1, 1, -1, -2, 1, -4, 0, -3, 1, -13, -4, 2, -4, 1, -46, -10, -5, 5, -5, 1, -166, -36, -6, -8, 9, -6, 1, -610, -126, -28, 0, -14, 14, -7, 1, -2269, -456, -92, -24, 10, -24, 20, -8, 1, -8518, -1674, -333, -63, -27, 27, -39, 27, -9, 1

Offset

0,5

Links

G. C. Greubel, Rows n = 0..100 of triangle, flattened

Formula

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.

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.

(-1)^(n+1) * Sum_{k=0..n} T(n,k) = 2*A181933(n).

T(n,0) = -A026641(n-1), n>0.

Example

Triangle begins as:
    0;
   -1,   1;
   -1,  -2,  1;
   -4,   0, -3,  1;
  -13,  -4,  2, -4,  1;
  -46, -10, -5,  5, -5,  1;

Mathematica

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 *)

Prog

(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);
(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
(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
(Sage)
def T(n, k):
  if (k==n==0): return 0
  elif (k==n): return 1
  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))
[[T(n, k) for k in (0..n)] for n in (0..10)] # G. C. Greubel, Apr 29 2019

Crossrefs

Sequence in context: A341509 A048614 A001442 * A158285 A277994 A334112

Adjacent sequences: A226949 A226950 A226951 * A226953 A226954 A226955

Keyword

sign,tabl

Author

Dmitry Kruchinin, Jun 24 2013

Status

approved