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

0,4

Comments

Riordan array (1/sqrt((1+x-x^2)(1-3x-x^2)), (1-x-x^2-sqrt((1+x-x^2)(1-3x-x^2))/(2x)). 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.

Links

Table of n, a(n) for n=0..61.

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)}}.

Example

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

Sequence in context: A016691 A177347 A101020 * A208612 A183157 A211957

Adjacent sequences: A160902 A160903 A160904 * A160906 A160907 A160908

Keyword

easy,nonn,tabl

Author

Paul Barry, May 29 2009

Status

approved