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 A117418 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). 11
 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, 208, 191, 79, 28, 7, 1, 1, 724, 723, 780, 322, 126, 37, 8, 1, 1, 2722, 2721, 3415, 1385, 572, 180, 48, 9, 1, 1, 11054, 11053, 15924, 6293, 2692, 871, 264, 58, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Here SHIFT_RIGHT(T) shifts the columns of T one place to the right and fills column 0 with [1,0,0,0,...]. LINKS G. C. Greubel, Rows n = 0..50 of the triangle, flattened FORMULA 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. EXAMPLE Triangle T begins: 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, 208, 191, 79, 28, 7, 1; 1, 724, 723, 780, 322, 126, 37, 8, 1; 1, 2722, 2721, 3415, 1385, 572, 180, 48, 9, 1; 1, 11054, 11053, 15924, 6293, 2692, 871, 264, 58, 10, 1; The matrix square T^2 = A117427: 1; 2, 1; 4, 4, 1; 9, 14, 6, 1; 23, 50, 28, 8, 1; 66, 191, 126, 48, 10, 1; 209, 780, 572, 264, 70, 12, 1; where column k of T^2 equals column 2k+1 of T. Let matrix R = SHIFT_RIGHT(T): 1; 0, 1; 0, 1, 1; 0, 1, 2, 1; 0, 1, 4, 3, 1; 0, 1, 9, 8, 4, 1; 0, 1, 23, 22, 14, 5, 1; then matrix product T*R = A117425: 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; where column k of T*R equals column 2k of T. MATHEMATICA 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}] ]]]; Table[A117418[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, May 31 2021 *) PROG (PARI) T(n, k)=if(nn): return 0 elif (k==0 or k==n): return 1 elif (k==n-1): return n 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)) flatten([[A117418(n, k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, May 31 2021 CROSSREFS Cf. A117419, A117420, A117421, A117422, A117423, A117424, A117425 (T*SHIFT_RIGHT(T)), A117427 (T^2), A117428. Sequence in context: A204849 A105632 A091491 * A101494 A125781 A091150 Adjacent sequences: A117415 A117416 A117417 * A117419 A117420 A117421 KEYWORD nonn,tabl AUTHOR Paul D. Hanna, Mar 14 2006 STATUS approved

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Last modified March 4 14:06 EST 2024. Contains 370532 sequences. (Running on oeis4.)