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Column 1 of triangle A117418; also row sums of triangle A117418; also column 0 of the matrix square of triangle A117418.
7

%I #6 May 31 2021 21:45:29

%S 1,2,4,9,23,66,209,724,2722,11054,48221,224549,1109099,5778542,

%T 31599830,180605495,1074867618,6641877098,42500291654,280917956734,

%U 1913770121847,13412187670520,96528097452230,712332226596239,5382561715509704

%N Column 1 of triangle A117418; also row sums of triangle A117418; also column 0 of the matrix square of triangle A117418.

%H G. C. Greubel, <a href="/A117419/b117419.txt">Table of n, a(n) for n = 0..650</a>

%F a(n) = A117418(n+1, 1). - _G. C. Greubel_, May 31 2021

%t A117418[n_, k_]:= A117418[n, k]= If[k<0 || k>n, 0, If[k==0 || k==n, 1, If[k==n-1, n, Sum[A117418[n -Floor[(k+1)/2], Floor[k/2] +j]*A117418[Floor[(k-1)/2] +j, Floor[(k-1)/2]], {j,0,n-k}] ]]];

%t Table[A117418[n+1,1], {n,0,30}] (* _G. C. Greubel_, May 31 2021 *)

%o (Sage)

%o @CachedFunction

%o def A117418(n, k):

%o if (k==0 or k==n): return 1

%o elif (k==n-1): return n

%o else: return sum( A117418(n -(k+1)//2, k//2 +j)*A117418((k-1)//2 +j, (k-1)//2) for j in (0..n-k))

%o [A117418(n+1, 1) for n in (0..30)] # _G. C. Greubel_, May 31 2021

%Y Cf. A117418, A117420, A117421, A117422, A117423, A117424.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 14 2006