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%I #13 Feb 21 2024 22:48:42
%S 1,1,1,1,3,1,1,8,7,1,1,25,44,15,1,1,111,346,208,31,1,1,809,4045,3720,
%T 912,63,1,1,10360,77351,99776,35136,3840,127,1,1,236952,2535715,
%U 4341249,2032888,308976,15808,255,1,1,9708797,145895764,319822055,189724354,37329584,2608864,64256,511,1
%N Lower triangular matrix T, read by rows, such that T(n,0) = 1 and T(n,k) = T(n-1,k) + T^2(n-1,k-1) for k>0, where T^2 is the matrix square of T.
%C This triangle has the same row sums and first column terms as in rows 2^n, for n>=0, of triangle A093662.
%H G. C. Greubel, <a href="/A097712/b097712.txt">Rows n = 0..50 of the triangle, flattened</a>
%F T(n, k) = T(n-1, k) + Sum_{j=0..n-1} T(n-1, j)*T(j, k-1), with T(n, 0) = T(n, n) = 1.
%F T(n, 1) = A097713(n-1), n >= 1.
%F Sum_{k=0..n} T(n, k) = A016121(n) (row sums).
%e T(5,1) = T(4,1) + T^2(4,0) = 25 + 86 = 111.
%e T(5,2) = T(4,2) + T^2(4,1) = 44 + 302 = 346.
%e T(5,3) = T(4,3) + T^2(4,2) = 15 + 193 = 208.
%e Rows of T begin:
%e 1;
%e 1, 1;
%e 1, 3, 1;
%e 1, 8, 7, 1;
%e 1, 25, 44, 15, 1;
%e 1, 111, 346, 208, 31, 1;
%e 1, 809, 4045, 3720, 912, 63, 1;
%e 1, 10360, 77351, 99776, 35136, 3840, 127, 1;
%e 1, 236952, 2535715, 4341249, 2032888, 308976, 15808, 255, 1;
%e Rows of T^2 begin:
%e 1;
%e 2, 1;
%e 5, 6, 1;
%e 17, 37, 14, 1;
%e 86, 302, 193, 30, 1;
%e 698, 3699, 3512, 881, 62, 1;
%e 9551, 73306, 96056, 34224, 3777, 126, 1;
%e 226592, 2458364, 4241473, 1997752, 305136, 15681, 254, 1;
%e Column 0 of T^2 forms A016121.
%e Row sums of T^2 form the first differences of A016121.
%t T[n_, k_] := T[n, k] = If[n < 0 || k > n, 0, If[n == k, 1, If[k == 0, 1, T[n - 1, k] + Sum[T[n - 1, j] T[j, k - 1], {j, 0, n - 1}]]]];
%t Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Oct 02 2019 *)
%o (PARI) T(n,k)=if(n<0 || k>n,0,if(n==k,1,if(k==0,1, T(n-1,k)+sum(j=0,n-1,T(n-1,j)*T(j,k-1));)))
%o (SageMath)
%o @CachedFunction
%o def T(n,k): # T = A097712
%o if k<0 or k>n: return 0
%o elif k==0 or k==n: return 1
%o else: return T(n-1,k) + sum(T(n-1,j)*T(j,k-1) for j in range(n))
%o flatten([[T(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Feb 20 2024
%Y Cf. A016121 (row sums), A093662, A097710, A097713.
%K nonn,tabl
%O 0,5
%A _Paul D. Hanna_, Aug 24 2004
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