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%I #7 May 31 2021 03:58:29
%S 1,1,1,1,3,1,1,8,5,1,1,22,20,7,1,1,65,79,37,9,1,1,208,322,180,58,11,1,
%T 1,723,1385,871,339,83,13,1,1,2721,6293,4296,1935,550,113,15,1,1,
%U 11053,30152,21821,11092,3465,846,148,17,1,1,48220,151842,114676,64748,21514,5911,1220,186,19,1
%N Triangle T, read by rows, formed by a column bisection of triangle A117418: column k of T equals column 2*k of A117418.
%H G. C. Greubel, <a href="/A117425/b117425.txt">Rows n = 0.50 of the triangle, flattened</a>
%F T(n, k) = A117418(n+k, 2*k). - _G. C. Greubel_, May 31 2021
%e Triangle begins:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 8, 5, 1;
%e 1, 22, 20, 7, 1;
%e 1, 65, 79, 37, 9, 1;
%e 1, 208, 322, 180, 58, 11, 1;
%e 1, 723, 1385, 871, 339, 83, 13, 1;
%e 1, 2721, 6293, 4296, 1935, 550, 113, 15, 1;
%e 1, 11053, 30152, 21821, 11092, 3465, 846, 148, 17, 1;
%e Column k of T equals column 2*k of A117418, which begins:
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 4, 3, 1;
%e 1, 9, 8, 4, 1;
%e 1, 23, 22, 14, 5, 1;
%e 1, 66, 65, 50, 20, 6, 1;
%e 1, 209, 208, 191, 79, 28, 7, 1;
%e Let matrix R = SHIFT_RIGHT(A117418):
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 1, 2, 1;
%e 0, 1, 4, 3, 1;
%e 0, 1, 9, 8, 4, 1;
%e 0, 1, 23, 22, 14, 5, 1;
%e 0, 1, 66, 65, 50, 20, 6, 1;
%e then the matrix product A117418*R yields this triangle.
%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 A117425[n_, k_]:= A117418[n+k, 2*k];
%t Table[A117425[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 31 2021 *)
%o (PARI)
%o A117418(n, k) = if(n<k || k<0, 0, if(n==k || k==0, 1, if(n==k+1, n, sum(j=0, n-k, A117418(n-((k+1)\2), k\2+j)*A117418((k-1)\2+j, (k-1)\2)))));
%o A117425(n, k) = A117418(n+k, 2*k);
%o for(n=0, 12, for(k=0,n, print1(A117425(n,k), ", "))) \\ modified by _G. C. Greubel_, May 31 2021
%o (Sage)
%o @CachedFunction
%o def A117418(n, k):
%o if (k<0 or k>n): return 0
%o elif (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 def A117425(n,k): return A117418(n+k,2*k)
%o flatten([[A117425(n, k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 31 2021
%Y Cf. A117418, A117426 (row sums), A117427 (dual).
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Mar 14 2006
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