A121908 S-D transform of Catalan numbers A000108.

1, 2, 3, 9, 19, 72, 181, 752, 2051, 8902, 25417, 113249, 333101, 1510888, 4538219, 20853973, 63626003, 295288350, 911918665, 4265460227, 13300767273, 62608960656, 196778953279, 931129725342, 2945833819213, 14000655099890, 44541071348599, 212484364171847

Offset: 0

Author

Philippe Deléham, Sep 01 2006

Keywords

• nonn

Examples

			1 1 2 5 14 42 132 ... (A000108)
2 1 7 9 56 90 ...
3 6 16 47 146 ...
9 10 63 99 ...
19 53 162 ...
72 109 ...
181 ...
Row 1 : A000108
Row 2 : 1+1=2, 2-1=1, 5+2=7, 14-5=9, 42+14=56, 132-42=90, ...
Row 3 : 1+2=3, 7-1=6, 9+7=16, 56-9=47, 90+56=146, ...
Row 4 : 6+3=9, 16-6=10, 47+16=63, 146-47=99, ...
Row 5 : 10+9=19, 63-10=53, 99+63=162, ...
Row 6 : 53+19=72, 162-53=109, ...
Row 7 : 109+72=181, ...
First diagonal of this triangular array form this sequence.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Programs

Maple

a:= proc(n) option remember; `if`(n<6, [1, 2, 3, 9, 19, 72][n+1],
      ((16*n^2+72*n-153)*n *a(n-1)
       +(304*n^4-1276*n^3+1213*n^2+487*n-754) *a(n-2)
       -(288*n^3-768*n^2-294*n+1424) *a(n-3)
       -(560*n^4-3772*n^3+6497*n^2+1253*n-4558) *a(n-4)
       +17*(n-4)*(16*n^2-8*n-29) *a(n-5)
       +17*(n-5)*(n-4)*(16*n^2-4*n-13) *a(n-6)) /
      (n*(n+1)*(16*n^2-36*n+7)))
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Jul 12 2014

Mathematica

T[n_, k_] := Binomial[Mod[n, 2], Mod[k, 2]] Binomial[Quotient[n, 2], Quotient[k, 2]];
a[n_] := Sum[T[n, k] CatalanNumber[k], {k, 0, n}];
a /@ Range[0, 40] (* Jean-François Alcover, Nov 19 2020 *)

Formula

a(n) = Sum_{k=0..n} A051159(n,k) * A000108(k).

Recurrence: see Maple program.

Extensions

More terms from Alois P. Heinz, Jul 12 2014