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%I #7 Sep 08 2022 08:45:41
%S 1,1,1,1,4,1,1,11,11,1,1,26,14608,26,1,1,57,152637,152637,57,1,1,120,
%T 1479726,251732291184,1479726,120,1,1,247,13824739,16871482830550,
%U 16871482830550,13824739,247,1,1,502,126781020,1103881308184906,113909683214485984529600,1103881308184906,126781020,502,1
%N 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.
%H G. C. Greubel, <a href="/A157221/b157221.txt">Rows n = 0..25 of the triangle, flattened</a>
%F 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.
%e Triangle begins as:
%e 1;
%e 1, 1;
%e 1, 4, 1;
%e 1, 11, 11, 1;
%e 1, 26, 14608, 26, 1;
%e 1, 57, 152637, 152637, 57, 1;
%e 1, 120, 1479726, 251732291184, 1479726, 120, 1;
%e 1, 247, 13824739, 16871482830550, 16871482830550, 13824739, 247, 1;
%t f[n_, k_]:= If[k<=Floor[n/2], k, n-k];
%t Eulerian[n_, k_]:= Sum[(-1)^j*Binomial[n+1,j]*(k+1-j)^n, {j,0,k+1}];
%t T[n_, k_]:= Eulerian[n*f[n,k] + 1, f[n,k]];
%t Table[Eulerian[n*f[n, k] +1, f[n, k]], {n,0,10}, {k,0,n}]//Flatten
%o (Magma)
%o f:= func< n,k | k le Floor(n/2) select k else n-k >;
%o Eulerian:= func< n,k | (&+[(-1)^j*Binomial(n+1,j)*(k-j+1)^n: j in [0..k+1]]) >;
%o [Eulerian(n*f(n,k)+1, f(n,k)): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jan 10 2022
%o (Sage)
%o def f(n,k): return k if (k <= (n//2)) else n-k
%o def Eulerian(n,k): return sum((-1)^j*binomial(n+1,j)*(k-j+1)^n for j in (0..k+1))
%o 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
%Y Cf. A008292, A157219.
%K nonn,tabl
%O 0,5
%A _Roger L. Bagula_, Feb 25 2009
%E Edited by _G. C. Greubel_, Jan 10 2022
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