A157117 Triangle T(n, k) = f(n, k) + f(n, n-k), where f(n, k) = A008292(n*k + 1, n-k) if k <= n otherwise A008292(n*(n-k), k), read by rows.
2, 1, 1, 1, 8, 1, 1, 131, 131, 1, 1, 8204, 29216, 8204, 1, 1, 2097187, 44136233, 44136233, 2097187, 1, 1, 2147483736, 846839476071, 503464582368, 846839476071, 2147483736, 1, 1, 8796093022411, 150092195453359483, 288159195861579519, 288159195861579519, 150092195453359483, 8796093022411, 1
Offset: 0
Examples
2; 1, 1; 1, 8, 1; 1, 131, 131, 1; 1, 8204, 29216, 8204, 1; 1, 2097187, 44136233, 44136233, 2097187, 1; 1, 2147483736, 846839476071, 503464582368, 846839476071, 2147483736, 1;
Links
- G. C. Greubel, Rows n = 0..25 of the triangle, flattened
Programs
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Magma
Eulerian:= func< n,k | (&+[(-1)^j*Binomial(n+1,j)*(k-j+1)^n: j in [0..k+1]]) >; f:= func< n,k | k le n select Eulerian(n*k+1,n-k) else Eulerian(n*(n-k)+1, k) >; A157117:= func< n,k | f(n,k) + f(n,n-k) >; [A157117(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 11 2022
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Mathematica
f[n_, k_]:= If[k<=n, Eulerian[n*k+1,n-k], Eulerian[n*(n-k)+1,k]]; T[n_, k_]:= f[n,k] + f[n,n-k]; Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jan 11 2022 *) -
Sage
def Eulerian(n,k): return sum((-1)^j*binomial(n+1,j)*(k-j+1)^n for j in (0..k+1)) def f(n,k): return Eulerian(n*k+1,n-k) if (k
Formula
Extensions
Edited by G. C. Greubel, Jan 11 2022