A117418 - OEIS

%I #12 May 31 2021 21:51:15

%S 1,1,1,1,2,1,1,4,3,1,1,9,8,4,1,1,23,22,14,5,1,1,66,65,50,20,6,1,1,209,

%T 208,191,79,28,7,1,1,724,723,780,322,126,37,8,1,1,2722,2721,3415,1385,

%U 572,180,48,9,1,1,11054,11053,15924,6293,2692,871,264,58,10,1

%N Triangle T, read by rows, such that column 2k+1 of T equals column k of T^2 and column 2k of T equals column k of T*R: [T^2](n+k,k) = T(n+2k+1,2k+1) and [T*R](n+k,k) = T(n+2k,2k) for n>=0, k>=0, where R = SHIFT_RIGHT(T).

%C Here SHIFT_RIGHT(T) shifts the columns of T one place to the right and fills column 0 with [1,0,0,0,...].

%H G. C. Greubel, <a href="/A117418/b117418.txt">Rows n = 0..50 of the triangle, flattened</a>

%F T(n,2k+1) = Sum_{j=0..n-2k-1} T(n-k-1,k+j)*T(k+j,k) for n>2k and T(n,2k) = Sum_{j=0..n-2k} T(n-k,k+j)*T(k-1+j,k-1) for n>=2k, with T(n,n) = T(n,0) = 1.

%e Triangle T 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   1,   724,   723,   780,  322,  126,  37,   8,  1;

%e   1,  2722,  2721,  3415, 1385,  572, 180,  48,  9,  1;

%e   1, 11054, 11053, 15924, 6293, 2692, 871, 264, 58, 10, 1;

%e The matrix square T^2 = A117427:

%e     1;

%e     2,   1;

%e     4,   4,   1;

%e     9,  14,   6,   1;

%e    23,  50,  28,   8,  1;

%e    66, 191, 126,  48, 10,  1;

%e   209, 780, 572, 264, 70, 12, 1;

%e where column k of T^2 equals column 2k+1 of T.

%e Let matrix R = SHIFT_RIGHT(T):

%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 then matrix product T*R = A117425:

%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 where column k of T*R equals column 2k of T.

%t A117418[n_, k_]:= T[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, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, May 31 2021 *)

%o (PARI) T(n,k)=if(n<k || k<0,0,if(n==k || k==0,1,if(n==k+1,n, sum(j=0,n-k,T(n-((k+1)\2),k\2+j)*T((k-1)\2+j,(k-1)\2)))))

%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 flatten([[A117418(n, k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, May 31 2021

%Y Cf. A117419, A117420, A117421, A117422, A117423, A117424, A117425 (T*SHIFT_RIGHT(T)), A117427 (T^2), A117428.

%K nonn,tabl

%O 0,5

%A _Paul D. Hanna_, Mar 14 2006