A275327 Triangle read by rows, Riordan array (1, (2+(x-1)/(2*x^2)*(1-sqrt(1-4*x^2)))/ sqrt(1-4*x^2)).

1, 0, 1, 0, 1, 1, 0, 3, 2, 1, 0, 2, 7, 3, 1, 0, 10, 10, 12, 4, 1, 0, 5, 33, 25, 18, 5, 1, 0, 35, 42, 78, 48, 25, 6, 1, 0, 14, 144, 144, 155, 80, 33, 7, 1, 0, 126, 168, 420, 356, 275, 122, 42, 8, 1, 0, 42, 610, 723, 1018, 736, 450, 175, 52, 9, 1

Offset: 0

Author

Peter Luschny, Aug 16 2016

Keywords

• nonn
• tabl

Examples

			Table starts:
[n] [k=0,1,2,...] row sum
[0] [1] 1
[1] [0, 1] 1
[2] [0, 1, 1] 2
[3] [0, 3, 2, 1] 6
[4] [0, 2, 7, 3, 1] 13
[5] [0, 10, 10, 12, 4, 1] 37
[6] [0, 5, 33, 25, 18, 5, 1] 87
[7] [0, 35, 42, 78, 48, 25, 6, 1] 235
[8] [0, 14, 144, 144, 155, 80, 33, 7, 1] 578
[9] [0, 126, 168, 420, 356, 275, 122, 42, 8, 1] 1518
		

Crossrefs

Cf. A057977 (column 1), A128899, A275328.

Programs

Maple

S := proc(n, k) option remember; local ecn:
if n = 0 then return n^k fi;
ecn := n -> n!/(iquo(n,2)!^2)/(iquo(n,2)+1);
add(ecn(i)*S(n-1,k-i), i=1..k-n+1) end:
A275327 := (n, k) -> S(k, n):
seq(seq(A275327(n, k),k=0..n),n=0..8);

Mathematica

(* The function RiordanArray is defined in A256893. *)
RiordanArray[1&, (2+(#-1)/(2#^2) (1-Sqrt[1-4#^2]))/Sqrt[1-4#^2]&, 11] // Flatten (* Jean-François Alcover, Jul 16 2019 *)

Sage

# uses[riordan_array from A256893]
s = (2+(x-1)/(2*x^2)*(1-sqrt(1-4*x^2)))/sqrt(1-4*x^2)
riordan_array(1, s, 12)