A160905 Right hand side of Pascal rhombus A059317.

1, 1, 1, 4, 2, 1, 9, 8, 3, 1, 29, 22, 13, 4, 1, 82, 72, 42, 19, 5, 1, 255, 218, 146, 70, 26, 6, 1, 773, 691, 476, 261, 107, 34, 7, 1, 2410, 2158, 1574, 914, 428, 154, 43, 8, 1, 7499, 6833, 5122, 3177, 1603, 659, 212, 53, 9, 1, 23575, 21612, 16706, 10816, 5867, 2628, 967

Offset: 0

Author

Paul Barry, May 29 2009

Keywords

• easy
• nonn
• tabl

Comments

Riordan array (1/sqrt((1+x-x^2)*(1-3*x-x^2)), (1-x-x^2-sqrt((1+x-x^2)*(1-3*x-x^2)))/(2*x)). Can be factored as

(1/(1-x-x^2), x/(1-x-x^2))*(1/sqrt(1-4x^2),xc(x^2)) = (1/(1-x^2),x/(1-x^2))*(1/(1-x),x/(1-x))*(1/sqrt(1-4x^2),xc(x^2))

and (1/(1-x^2),x/(1-x^2))*(1/sqrt(1-2x-3x^2),(1-x-sqrt(1-2x-3x^2))/(2x)).

Here, c(x) is the g.f. of the Catalan numbers A000108.

A160905 = A037027*A108044 = abs(A049310)*A007318*A108044 = abs(A049310)*A094531.

Examples

			Triangle begins:
    1;
    1,   1;
    4,   2,   1;
    9,   8,   3,  1;
   29,  22,  13,  4,  1;
   82,  72,  42, 19,  5, 1;
  255, 218, 146, 70, 26, 6, 1;
  ...
		

Crossrefs

Left column gives A059345.

Cf. A059317.

Formula

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

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