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

0,2

Links

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

Formula

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

Recurrence: see Maple program.

Example

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.

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 *)

Crossrefs

Sequence in context: A324374 A106519 A006866 * A231368 A369517 A245123

Adjacent sequences: A121905 A121906 A121907 * A121909 A121910 A121911

Keyword

nonn

Author

Philippe Deléham, Sep 01 2006

Extensions

More terms from Alois P. Heinz, Jul 12 2014

Status

approved