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A225413
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Triangle read by rows: T(n,k) = (A101164(n,k) - A014473(n,k))/2.
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2
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0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 3, 3, 0, 0, 0, 0, 6, 12, 6, 0, 0, 0, 0, 10, 30, 30, 10, 0, 0, 0, 0, 15, 60, 91, 60, 15, 0, 0, 0, 0, 21, 105, 215, 215, 105, 21, 0, 0, 0, 0, 28, 168, 435, 590, 435, 168, 28, 0, 0, 0, 0, 36, 252, 791, 1365, 1365, 791, 252, 36, 0, 0
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,18
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COMMENTS
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LINKS
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FORMULA
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T(n, n-k) = T(n, k).
Sum_{k=0..n} T(n, k) = (A000129(n+1) + n + 1 - 2^(n+1))/2.
Sum_{k=0..n} (-1)^k*T(n, k) = A121262(n) - [n=0]. (End)
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EXAMPLE
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Triangle begins as:
0;
0, 0;
0, 0, 0;
0, 0, 0, 0;
0, 0, 1, 0, 0;
0, 0, 3, 3, 0, 0;
0, 0, 6, 12, 6, 0, 0;
0, 0, 10, 30, 30, 10, 0, 0;
0, 0, 15, 60, 91, 60, 15, 0, 0;
0, 0, 21, 105, 215, 215, 105, 21, 0, 0;
0, 0, 28, 168, 435, 590, 435, 168, 28, 0, 0;
0, 0, 36, 252, 791, 1365, 1365, 791, 252, 36, 0, 0;
0, 0, 45, 360, 1330, 2800, 3571, 2800, 1330, 360, 45, 0, 0;
0, 0, 55, 495, 2106, 5250, 8197, 8197, 5250, 2106, 495, 55, 0, 0;
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MATHEMATICA
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T[n_, k_]:= ((-1)^(n-k)*Hypergeometric2F1[-n+k, k+1, 1, 2] - 2*Binomial[n, k] +1)/2;
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Apr 08 2024 *)
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PROG
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(Haskell)
a225413 n k = a225413_tabl !! n !! k
a225413_row n = a225413_tabl !! n
a225413_tabl = map (map (`div` 2)) $
zipWith (zipWith (-)) a101164_tabl a014473_tabl
(Magma)
A008288:= func< n, k | (&+[Binomial(n-j, j)*Binomial(n-2*j, k-j): j in [0..k]]) >;
(SageMath)
def A008288(n, k): return sum(binomial(n-j, j)*binomial(n-2*j, k-j) for j in range(k+1))
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CROSSREFS
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3rd column = A000217 (triangular numbers).
4th column = A027480 (n(n+1)(n+2)/2).
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KEYWORD
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AUTHOR
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STATUS
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approved
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