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A097712 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. 10
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,5
COMMENTS
This triangle has the same row sums and first column terms as in rows 2^n, for n>=0, of triangle A093662.
LINKS
FORMULA
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.
T(n, 1) = A097713(n-1), n >= 1.
Sum_{k=0..n} T(n, k) = A016121(n) (row sums).
EXAMPLE
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;
Column 0 of T^2 forms A016121.
Row sums of T^2 form the first differences of A016121.
MATHEMATICA
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}]]]];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 02 2019 *)
PROG
(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
def T(n, k): # T = A097712
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
CROSSREFS
Cf. A016121 (row sums), A093662, A097710, A097713.
Sequence in context: A263859 A124469 A094816 * A238688 A174117 A157210
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Aug 24 2004
STATUS
approved

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)