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A174303 A symmetrical triangle: T(n,k) = A008292(n+1, k) * f(n,k), where f(n,k) = 2^k when floor(n/2) >= k, otherwise 2^(n-k).

%I A174303 #9 Feb 16 2025 08:33:12
%S A174303 1,1,1,1,8,1,1,22,22,1,1,52,264,52,1,1,114,1208,1208,114,1,1,240,4764,
%T A174303 19328,4764,240,1,1,494,17172,124952,124952,17172,494,1,1,1004,58432,
%U A174303 705872,2499040,705872,58432,1004,1,1,2026,191360,3641536,20965664,20965664,3641536,191360,2026,1
%N A174303 A symmetrical triangle: T(n,k) = A008292(n+1, k) * f(n,k), where f(n,k) = 2^k when floor(n/2) >= k, otherwise 2^(n-k).
%C A174303 Row sums are: {1, 2, 10, 46, 370, 2646, 29338, 285238, 4029658, ...}.
%H A174303 G. C. Greubel, <a href="/A174303/b174303.txt">Rows n = 0..100 of triangle, flattened</a>
%H A174303 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/EulerianNumber.html">Eulerian Number</a>
%F A174303 T(n,k) = Eulerian(n+1, k)*if(floor(n/2) greater than or equal to k then 2^m otherwise 2^(n-k)), where the Eulerian numbers are defined as A008292(n,k).
%e A174303 Triangle begins as:
%e A174303   1;
%e A174303   1,    1;
%e A174303   1,    8,     1;
%e A174303   1,   22,    22,      1;
%e A174303   1,   52,   264,     52,       1;
%e A174303   1,  114,  1208,   1208,     114,      1;
%e A174303   1,  240,  4764,  19328,    4764,    240,     1;
%e A174303   1,  494, 17172, 124952,  124952,  17172,   494,    1;
%e A174303   1, 1004, 58432, 705872, 2499040, 705872, 58432, 1004,   1;
%t A174303 Eulerian[n_, k_]:= Sum[(-1)^j*Binomial[n+1,j]*(k-j+1)^n, {j,0,k+1}];
%t A174303 Table[Eulerian[n+1,m]*If[Floor[n/2] >= m, 2^m, 2^(n-m)], {n,0,10}, {m,0,n} ]//Flatten (* modified by _G. C. Greubel_, Apr 15 2019 *)
%o A174303 (PARI) {eulerian(n,k) = sum(j=0,k+1, (-1)^j*binomial(n+1,j)*(k-j+1)^n)};
%o A174303 for(n=0,10, for(k=0,n, print1(eulerian(n+1,k)*if(floor(n/2)>=k, 2^k, 2^(n-k)), ", "))) \\ _G. C. Greubel_, Apr 15 2019
%o A174303 (Magma) Eulerian:= func< n,k | (&+[(-1)^j*Binomial(n+1,j)*(k-j+1)^n: j in [0..k+1]]) >; [[Floor(n/2) ge k select 2^k*Eulerian(n+1,k) else 2^(n-k)*Eulerian(n+1,k): k in [0..n]]: n in [0..10]]; // _G. C. Greubel_, Apr 15 2019
%o A174303 (Sage)
%o A174303 def Eulerian(n,k): return sum((-1)^j*binomial(n+1,j)*(k-j+1)^n for j in (0..k+1))
%o A174303 def T(n,k):
%o A174303    if floor(n/2)>=k: return 2^k*Eulerian(n+1,k)
%o A174303    else: return 2^(n-k)*Eulerian(n+1,k)
%o A174303 [[T(n,k) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Apr 15 2019
%Y A174303 Cf. A008292, A144463, A144470, A174301.
%K A174303 nonn,tabl
%O A174303 0,5
%A A174303 _Roger L. Bagula_, Mar 15 2010
%E A174303 Edited by _G. C. Greubel_, Apr 15 2019