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A174098 Symmetrical triangle T(n, m) = floor(Eulerian(n+1, m)/2), read by rows.

%I A174098 #13 Sep 08 2022 08:45:51
%S A174098 2,5,5,13,33,13,28,151,151,28,60,595,1208,595,60,123,2146,7809,7809,
%T A174098 2146,123,251,7304,44117,78095,44117,7304,251,506,23920,227596,655177,
%U A174098 655177,227596,23920,506,1018,76318,1101744,4869057,7862124,4869057,1101744,76318,1018
%N A174098 Symmetrical triangle T(n, m) = floor(Eulerian(n+1, m)/2), read by rows.
%C A174098 Row sums are: {2, 10, 59, 358, 2518, 20156, 181439, 1814398, 19958398, 239500796, 3113510398, 43589145596, 653837183996, ...}.
%H A174098 G. C. Greubel, <a href="/A174098/b174098.txt">Rows n = 2..100 of triangle, flattened</a>
%F A174098 T(n, m) = floor(Eulerian(n+1, m)/2), where Eulerian(n,k) = A008292(n,k).
%e A174098 Triangle begins as:
%e A174098     2;
%e A174098     5,     5;
%e A174098    13,    33,     13;
%e A174098    28,   151,    151,     28;
%e A174098    60,   595,   1208,    595,     60;
%e A174098   123,  2146,   7809,   7809,   2146,    123;
%e A174098   251,  7304,  44117,  78095,  44117,   7304,   251;
%e A174098   506, 23920, 227596, 655177, 655177, 227596, 23920, 506;
%t A174098 Eulerian[n_, k_]:= Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j, 0, k+1}];
%t A174098 Table[Floor[Eulerian[n+1, m]/2], {n, 2, 12}, {m, 1, n-1}]//Flatten (* _G. C. Greubel_, Apr 25 2019 *)
%o A174098 (PARI) {T(n,k) = (sum(j=0,k+1, (-1)^j*binomial(n+2,j)*(k-j+1)^(n+1)))\2};
%o A174098 for(n=2,12, for(k=1,n-1, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Apr 25 2019
%o A174098 (Magma) [[Floor((&+[(-1)^j*Binomial(n+2,j)*(k-j+1)^(n+1): j in [0..k+1]] )/2): k in [1..n-1]]: n in [2..12]]; // _G. C. Greubel_, Apr 25 2019
%o A174098 (Sage) [[floor(sum((-1)^j*binomial(n+2,j)*(k-j+1)^(n+1) for j in (0..k+1))/2) for k in (1..n-1)] for n in (2..12)] # _G. C. Greubel_, Apr 25 2019
%Y A174098 Cf. A008292, A166454.
%K A174098 nonn,tabl
%O A174098 2,1
%A A174098 _Roger L. Bagula_, Mar 07 2010
%E A174098 Edited by _G. C. Greubel_, Apr 25 2019