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A097084
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Triangle, read by rows, where the n-th diagonal equals the n-th row transformed by triangle A008459 (squared binomial coefficients).
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2
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1, 1, 1, 1, 2, 1, 1, 3, 5, 1, 1, 4, 10, 10, 1, 1, 5, 18, 28, 17, 1, 1, 6, 27, 74, 69, 26, 1, 1, 7, 39, 137, 245, 151, 37, 1, 1, 8, 52, 236, 586, 676, 298, 50, 1, 1, 9, 68, 372, 1194, 2126, 1634, 540, 65, 1, 1, 10, 85, 552, 2322, 5152, 6620, 3578, 913, 82, 1, 1, 11, 105, 777, 3954, 12002, 19292, 18082, 7249, 1459, 101, 1
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OFFSET
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0,5
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COMMENTS
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LINKS
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FORMULA
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T(n,k) = Sum_{j=0..k} T(n-k,j)*C(k,j)^2.
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EXAMPLE
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T(8,3) = 236 = (1)*1^2 + (5)*3^2 + (18)*3^2 + (28)*1^2
= Sum_{j=0..3} T(5,j)*C(3,j)^2.
Rows begin:
[1],
[1,1],
[1,2,1],
[1,3,5,1],
[1,4,10,10,1],
[1,5,18,28,17,1],
[1,6,27,74,69,26,1],
[1,7,39,137,245,151,37,1],
[1,8,52,236,586,676,298,50,1],...
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MAPLE
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T:= proc(n, k) option remember;
`if`(n=k or k=0, 1, `if`(k<0 or k>n, 0,
add(T(n-k, j)*binomial(k, j)^2, j=0..k)))
end:
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MATHEMATICA
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T[_, 0] = 1; T[n_, n_] = 1; T[n_, k_] /; 0 < k < n := T[n, k] = Sum[T[n - k, j]*Binomial[k, j]^2, {j, 0, k}]; T[_, _] = 0;
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PROG
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(PARI) T(n, k)=if(n<k || k<0, 0, if(n==k || k==0, 1, sum(j=0, n-k, T(n-k, j)*binomial(k, j)^2)))
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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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