A116389 Riordan array (1/sqrt(1-4*x^2), (1+x)/sqrt(1-4*x^2) -1).

1, 0, 1, 2, 2, 1, 0, 4, 4, 1, 6, 10, 10, 6, 1, 0, 16, 28, 20, 8, 1, 20, 44, 62, 62, 34, 10, 1, 0, 64, 152, 168, 120, 52, 12, 1, 70, 186, 328, 436, 374, 210, 74, 14, 1, 0, 256, 748, 1084, 1072, 736, 340, 100, 16, 1, 252, 772, 1606, 2598, 2924, 2332, 1326, 518, 130, 18, 1

Offset: 0

Author

Paul Barry, Feb 12 2006

Keywords

• easy
• nonn
• tabl

Examples

			Triangle begins:
   1;
   0,  1;
   2,  2,  1;
   0,  4,  4,  1;
   6, 10, 10,  6,  1;
   0, 16, 28, 20,  8,  1;
  20, 44, 62, 62, 34, 10, 1;
		

Links

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

Crossrefs

Row sums are A116390. Diagonal sums are A116391.

Product of A007318 and this sequence is A116392.

Programs

Magma

[[(&+[ (&+[ Round((-1)^(k-j)*4^r*Binomial(k,j)*Binomial(j, n-2*r)*Gamma(r+(j+1)/2)/(Factorial(r)*Gamma((j+1)/2))) : r in [0..Floor(n/2)]]) : j in [0..k]]) : k in [0..n]]: n in [0..10]]; // G. C. Greubel, May 23 2019

Mathematica

T[n_,k_]:= Sum[(-1)^(k-j)*Binomial[k,j]*Sum[4^r*Binomial[r+(j-1)/2, r]* Binomial[j, n-2*r], {r,0,Floor[n/2]}], {j,0,k}]; Table[T[n,k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, May 23 2019 *)

PARI

{T(n,k) = sum(j=0,k, sum(r=0,floor(n/2), (-1)^(k-j)*4^r* binomial(k,j)*binomial(r+(j-1)/2, r)*binomial(j, n-2*r) ))}; \\ G. C. Greubel, May 23 2019

Sage

[[sum( sum( (-1)^(k-j)*4^r* binomial(k,j)*binomial(r+(j-1)/2, r)*binomial(j, n-2*r) for r in (0..floor(n/2))) for j in (0..k)) for k in (0..n)] for n in (0..10)] # G. C. Greubel, May 23 2019

Formula

T(n,k) = Sum_{j=0..k} (-1)^(k-j)*C(k,j) * Sum_{i=0..floor(n/2)} 4^i * C(i+(j-1)/2, i)*C(j,n-2*i).