A143019 Infinite square array read by antidiagonals: a(q,n) is the coefficient of z^n in the series expansion of C(z)^q/(1-4z)^(3/2), where C(z) = (1-sqrt(1-4z))/(2z) is the Catalan function (q,n = 0,1,2,...).

1, 1, 6, 1, 7, 30, 1, 8, 38, 140, 1, 9, 47, 187, 630, 1, 10, 57, 244, 874, 2772, 1, 11, 68, 312, 1186, 3958, 12012, 1, 12, 80, 392, 1578, 5536, 17548, 51480, 1, 13, 93, 485, 2063, 7599, 25147, 76627, 218790, 1, 14, 107, 592, 2655, 10254, 35401, 112028, 330818

Offset: 0

Author

Emeric Deutsch, Jul 24 2008

Keywords

• nonn
• tabl

Comments

a(q,n) = a(q-1,n) + a(q+1,n-1).

Row 0 is A002457; row 1 is A000531; row 2 is A029760; row 3 is A045720.

Examples

			Array starts:
  1, 6, 30, 140,  630, ...
  1, 7, 38, 187,  874, ...
  1, 8, 47, 244, 1186, ...
  1, 9, 57, 312, 1578, ...
  ...
		

Crossrefs

Cf. A002457, A000531, A029760, A045720.

Programs

Maple

a:=proc(q,n) options operator, arrow: sum(4^i*binomial(2*n-2*i+q, n-i), i= 0.. n) end proc: aa:=proc(q,n) options operator, arrow: a(q-1,n-1) end proc: matrix(10,10,aa); # yields sequence in matrix form

Formula

a(q,n) = Sum_{i=0..n} 4^i*binomial(2n-2i+q, n-i).