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A121908 S-D transform of Catalan numbers A000108.

%I A121908 #15 Nov 19 2020 11:59:16
%S A121908 1,2,3,9,19,72,181,752,2051,8902,25417,113249,333101,1510888,4538219,
%T A121908 20853973,63626003,295288350,911918665,4265460227,13300767273,
%U A121908 62608960656,196778953279,931129725342,2945833819213,14000655099890,44541071348599,212484364171847
%N A121908 S-D transform of Catalan numbers A000108.
%H A121908 Alois P. Heinz, <a href="/A121908/b121908.txt">Table of n, a(n) for n = 0..1000</a>
%F A121908 a(n) = Sum_{k=0..n} A051159(n,k) * A000108(k).
%F A121908 Recurrence: see Maple program.
%e A121908 1 1 2 5 14 42 132 ... (A000108)
%e A121908 2 1 7 9 56 90 ...
%e A121908 3 6 16 47 146 ...
%e A121908 9 10 63 99 ...
%e A121908 19 53 162 ...
%e A121908 72 109 ...
%e A121908 181 ...
%e A121908 Row 1 : A000108
%e A121908 Row 2 : 1+1=2, 2-1=1, 5+2=7, 14-5=9, 42+14=56, 132-42=90, ...
%e A121908 Row 3 : 1+2=3, 7-1=6, 9+7=16, 56-9=47, 90+56=146, ...
%e A121908 Row 4 : 6+3=9, 16-6=10, 47+16=63, 146-47=99, ...
%e A121908 Row 5 : 10+9=19, 63-10=53, 99+63=162, ...
%e A121908 Row 6 : 53+19=72, 162-53=109, ...
%e A121908 Row 7 : 109+72=181, ...
%e A121908 First diagonal of this triangular array form this sequence.
%p A121908 a:= proc(n) option remember; `if`(n<6, [1, 2, 3, 9, 19, 72][n+1],
%p A121908       ((16*n^2+72*n-153)*n *a(n-1)
%p A121908        +(304*n^4-1276*n^3+1213*n^2+487*n-754) *a(n-2)
%p A121908        -(288*n^3-768*n^2-294*n+1424) *a(n-3)
%p A121908        -(560*n^4-3772*n^3+6497*n^2+1253*n-4558) *a(n-4)
%p A121908        +17*(n-4)*(16*n^2-8*n-29) *a(n-5)
%p A121908        +17*(n-5)*(n-4)*(16*n^2-4*n-13) *a(n-6)) /
%p A121908       (n*(n+1)*(16*n^2-36*n+7)))
%p A121908     end:
%p A121908 seq(a(n), n=0..40);  # _Alois P. Heinz_, Jul 12 2014
%t A121908 T[n_, k_] := Binomial[Mod[n, 2], Mod[k, 2]] Binomial[Quotient[n, 2], Quotient[k, 2]];
%t A121908 a[n_] := Sum[T[n, k] CatalanNumber[k], {k, 0, n}];
%t A121908 a /@ Range[0, 40] (* _Jean-François Alcover_, Nov 19 2020 *)
%K A121908 nonn
%O A121908 0,2
%A A121908 _Philippe Deléham_, Sep 01 2006
%E A121908 More terms from _Alois P. Heinz_, Jul 12 2014