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A157221
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Triangle T(n, k) = Eulerian(n*f(n, k) + 1, f(n, k)), where f(n, k) = k if k <= floor(n/2) otherwise n-k, read by rows.
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2
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1, 1, 1, 1, 4, 1, 1, 11, 11, 1, 1, 26, 14608, 26, 1, 1, 57, 152637, 152637, 57, 1, 1, 120, 1479726, 251732291184, 1479726, 120, 1, 1, 247, 13824739, 16871482830550, 16871482830550, 13824739, 247, 1, 1, 502, 126781020, 1103881308184906, 113909683214485984529600, 1103881308184906, 126781020, 502, 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) = Eulerian(n*f(n, k) + 1, f(n, k)), where f(n, k) = k if k <= floor(n/2) otherwise n-k.
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EXAMPLE
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Triangle begins as:
1;
1, 1;
1, 4, 1;
1, 11, 11, 1;
1, 26, 14608, 26, 1;
1, 57, 152637, 152637, 57, 1;
1, 120, 1479726, 251732291184, 1479726, 120, 1;
1, 247, 13824739, 16871482830550, 16871482830550, 13824739, 247, 1;
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MATHEMATICA
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f[n_, k_]:= If[k<=Floor[n/2], k, n-k];
Eulerian[n_, k_]:= Sum[(-1)^j*Binomial[n+1, j]*(k+1-j)^n, {j, 0, k+1}];
T[n_, k_]:= Eulerian[n*f[n, k] + 1, f[n, k]];
Table[Eulerian[n*f[n, k] +1, f[n, k]], {n, 0, 10}, {k, 0, n}]//Flatten
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PROG
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(Magma)
f:= func< n, k | k le Floor(n/2) select k else n-k >;
Eulerian:= func< n, k | (&+[(-1)^j*Binomial(n+1, j)*(k-j+1)^n: j in [0..k+1]]) >;
[Eulerian(n*f(n, k)+1, f(n, k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 10 2022
(Sage)
def f(n, k): return k if (k <= (n//2)) else n-k
def Eulerian(n, k): return sum((-1)^j*binomial(n+1, j)*(k-j+1)^n for j in (0..k+1))
flatten([[Eulerian(n*f(n, k)+1, f(n, k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jan 10 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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