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%I #7 Sep 08 2022 08:45:41
%S 2,1,1,1,8,1,1,131,131,1,1,8204,29216,8204,1,1,2097187,44136233,
%T 44136233,2097187,1,1,2147483736,846839476071,503464582368,
%U 846839476071,2147483736,1,1,8796093022411,150092195453359483,288159195861579519,288159195861579519,150092195453359483,8796093022411,1
%N 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.
%H G. C. Greubel, <a href="/A157117/b157117.txt">Rows n = 0..25 of the triangle, flattened</a>
%F 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).
%F T(n, n-k) = T(n, k).
%e 2;
%e 1, 1;
%e 1, 8, 1;
%e 1, 131, 131, 1;
%e 1, 8204, 29216, 8204, 1;
%e 1, 2097187, 44136233, 44136233, 2097187, 1;
%e 1, 2147483736, 846839476071, 503464582368, 846839476071, 2147483736, 1;
%t f[n_, k_]:= If[k<=n, Eulerian[n*k+1,n-k], Eulerian[n*(n-k)+1,k]];
%t T[n_, k_]:= f[n,k] + f[n,n-k];
%t Table[T[n, k], {n,0,10}, {k,0,n}]//Flatten (* modified by _G. C. Greubel_, Jan 11 2022 *)
%o (Magma)
%o Eulerian:= func< n,k | (&+[(-1)^j*Binomial(n+1,j)*(k-j+1)^n: j in [0..k+1]]) >;
%o f:= func< n,k | k le n select Eulerian(n*k+1,n-k) else Eulerian(n*(n-k)+1, k) >;
%o A157117:= func< n,k | f(n,k) + f(n,n-k) >;
%o [A157117(n,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 11 2022
%o (Sage)
%o def Eulerian(n,k): return sum((-1)^j*binomial(n+1,j)*(k-j+1)^n for j in (0..k+1))
%o def f(n,k): return Eulerian(n*k+1,n-k) if (k<n+1) else Eulerian(n*(n-k)+1, k)
%o def A157117(n,k): return f(n,k) + f(n,n-k)
%o flatten([[A157117(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jan 11 2022
%Y Cf. A008292, A157114, A157118.
%K nonn,tabl
%O 0,1
%A _Roger L. Bagula_, Feb 23 2009
%E Edited by _G. C. Greubel_, Jan 11 2022
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