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A157243
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Triangle T(n, k) = A001263(n*f(n,k) + 1, f(n,k) + 1), where f(n, k) = k if k <= floor(n/2) otherwise n-k, read by rows.
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1
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1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 336, 10, 1, 1, 15, 825, 825, 15, 1, 1, 21, 1716, 197676, 1716, 21, 1, 1, 28, 3185, 512050, 512050, 3185, 28, 1, 1, 36, 5440, 1163800, 294296640, 1163800, 5440, 36, 1, 1, 45, 8721, 2395575, 778076145, 778076145, 2395575, 8721, 45, 1
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OFFSET
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0,5
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LINKS
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FORMULA
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T(n, k) = A001263(n*f(n,k) + 1, f(n,k) + 1), where f(n, k) = k if k <= floor(n/2) otherwise n-k.
T(n, n-k) = T(n, k)).
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EXAMPLE
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Triangle begins as:
1;
1, 1;
1, 3, 1;
1, 6, 6, 1;
1, 10, 336, 10, 1;
1, 15, 825, 825, 15, 1;
1, 21, 1716, 197676, 1716, 21, 1;
1, 28, 3185, 512050, 512050, 3185, 28, 1;
1, 36, 5440, 1163800, 294296640, 1163800, 5440, 36, 1;
1, 45, 8721, 2395575, 778076145, 778076145, 2395575, 8721, 45, 1;
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MATHEMATICA
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f[n_, k_]:= If[k<=Floor[n/2], k, n-k];
A001263[n_, k_]:= Binomial[n-1, k-1]*Binomial[n, k]/(n-k+1);
T[n_, k_]:= A001263[n*f[n, k] +1, f[n, k] +1];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Jan 11 2022 *)
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PROG
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(Magma)
f:= func< n, k | k le Floor(n/2) select k else n-k >;
A001263:= func< n, k | Binomial(n-1, k-1)*Binomial(n, k)/(n-k+1) >;
(Sage)
def f(n, k): return k if (k <= (n//2)) else n-k
def A001263(n, k): return binomial(n-1, k-1)*binomial(n, k)/(n-k+1)
flatten([[A001263(n*f(n, k)+1, f(n, k)+1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 11 2022
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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