A127543 Triangle T(n,k), 0<=k<=n, read by rows given by :[ -1,1,1,1,1,1,1,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938.

1, -1, 1, 0, -1, 1, -1, 1, -1, 1, -2, 0, 2, -1, 1, -6, 2, 1, 3, -1, 1, -18, 5, 7, 2, 4, -1, 1, -57, 17, 19, 13, 3, 5, -1, 1, -186, 56, 64, 36, 20, 4, 6, -1, 1, -622, 190, 212, 124, 56, 28, 5, 7, -1, 1, -2120, 654, 722, 416, 198, 79, 37, 6, 8, -1, 1, -7338, 2282, 2494, 1434, 673, 287, 105, 47, 7, 9, -1, 1

Offset: 0

Author

Philippe Deléham, Apr 01 2007

Keywords

• sign
• tabl

Comments

Riordan array (2/(3-sqrt(1-4*x)), (1-sqrt(1-4*x))/(3-sqrt(1-4*x))). - Philippe Deléham, Jan 27 2014

Examples

			Triangle begins:
    1;
   -1,  1;
    0, -1,  1;
   -1,  1, -1,  1;
   -2,  0,  2, -1,  1;
   -6,  2,  1,  3, -1,  1;
  -18,  5,  7,  2,  4, -1,  1;
  -57, 17, 19, 13,  3,  5, -1, 1;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Programs

Mathematica

A065600[n_, k_]:= If[k==n, 1, Sum[j*Binomial[k+j, j]*Binomial[2*(n-k-j), n-k]/(n-k-j), {j,0, Floor[(n-k)/2]}]];
A127543[n_, k_]:= A065600[n-1,k-1] - A065600[n-1,k];
Table[A127543[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, May 17 2021 *)

Sage

def A065600(n,k): return 1 if (k==n) else sum( j*binomial(k+j, j)*binomial(2*(n-k-j), n-k)/(n-k-j) for j in (0..(n-k)//2) )
def A127543(n,k): return A065600(n-1, k-1) - A065600(n-1, k)
flatten([[A127543(n,k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 17 2021

Formula

T(n,k) = A065600(n-1,k-1) - A065600(n-1,k).

Sum_{k=0..n} T(n,k)*x^k = A127053(n), A126985(n), A127016(n), A127017(n), A126987(n), A126986(n), A126982(n), A126984(n), A126983(n), A000007(n), A000108(n), A000984(n), A007854(n), A076035(n), A076036(n), A127628(n), A126694(n), A115970(n) for n= -8,-7,...,8,9 respectively.

Sum_{j>=0} T(n,j)*A007318(j,k) = A106566(n,k).

Sum_{j>=0} T(n,j)*A038207(j,k) = A039599(n,k).

Sum_{j>=0} T(n,j)*A027465(j,k) = A116395(n,k).