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A339000
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Triangle read by rows: T(n, k) = C(n, k)*Sum_{j=0..n} C(n, k-j)*C(n+j, j)/C(2*j, j).
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1
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1, 1, 2, 1, 7, 5, 1, 15, 32, 13, 1, 26, 111, 123, 34, 1, 40, 285, 603, 429, 89, 1, 57, 610, 2094, 2748, 1408, 233, 1, 77, 1155, 5845, 12170, 11196, 4437, 610, 1, 100, 2002, 14014, 42355, 60686, 42255, 13587, 1597, 1, 126, 3246, 30030, 124137, 254756, 271961, 150951, 40736, 4181
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,3
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LINKS
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FORMULA
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T(n,n) = Fibonacci(2*n+1).
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EXAMPLE
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Triangle begins as:
1;
1, 2;
1, 7, 5;
1, 15, 32, 13;
1, 26, 111, 123, 34;
1, 40, 285, 603, 429, 89;
1, 57, 610, 2094, 2748, 1408, 233;
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MATHEMATICA
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T[n_, k_]:= With[{B=Binomial}, B[n, k]*Sum[B[n, k-j]*B[n+j, j]/B[2*j, j], {j, 0, n}]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jul 31 2024 *)
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PROG
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(Maxima)
T(n, m):=(binomial(n, m))*sum(((binomial(n, m-k))*(binomial(n+k, k)) )/(binomial(2*k, k)), k, 0, n);
(Magma)
b:=Binomial;
A339000:= func< n, k | b(n, k)*(&+[b(n, k-j)*b(n+j, j)/b(2*j, j): j in [0..n]]) >;
(SageMath)
b=binomial
def A339000(n, k): return b(n, k)*sum(b(n, k-j)*b(n+j, j)//b(2*j, j) for j in range(n+1))
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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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