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A097712
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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.
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10
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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, 912, 63, 1, 1, 10360, 77351, 99776, 35136, 3840, 127, 1, 1, 236952, 2535715, 4341249, 2032888, 308976, 15808, 255, 1, 1, 9708797, 145895764, 319822055, 189724354, 37329584, 2608864, 64256, 511, 1
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OFFSET
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0,5
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COMMENTS
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This triangle has the same row sums and first column terms as in rows 2^n, for n>=0, of triangle A093662.
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LINKS
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FORMULA
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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.
Sum_{k=0..n} T(n, k) = A016121(n) (row sums).
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EXAMPLE
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T(5,1) = T(4,1) + T^2(4,0) = 25 + 86 = 111.
T(5,2) = T(4,2) + T^2(4,1) = 44 + 302 = 346.
T(5,3) = T(4,3) + T^2(4,2) = 15 + 193 = 208.
Rows of T begin:
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, 912, 63, 1;
1, 10360, 77351, 99776, 35136, 3840, 127, 1;
1, 236952, 2535715, 4341249, 2032888, 308976, 15808, 255, 1;
Rows of T^2 begin:
1;
2, 1;
5, 6, 1;
17, 37, 14, 1;
86, 302, 193, 30, 1;
698, 3699, 3512, 881, 62, 1;
9551, 73306, 96056, 34224, 3777, 126, 1;
226592, 2458364, 4241473, 1997752, 305136, 15681, 254, 1;
Row sums of T^2 form the first differences of A016121.
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MATHEMATICA
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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}]]]];
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PROG
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(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)); )))
(SageMath)
@CachedFunction
if k<0 or k>n: return 0
elif k==0 or k==n: return 1
else: return T(n-1, k) + sum(T(n-1, j)*T(j, k-1) for j in range(n))
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Feb 20 2024
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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