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A156141 Triangle T(n,k) = A008517(n,k+1)+A008517(n,n+1-k) read by rows.

%I A156141 #4 Feb 16 2025 08:33:09
%S A156141 2,1,1,1,4,1,1,14,14,1,1,46,116,46,1,1,172,772,772,172,1,1,834,5160,
%T A156141 8800,5160,834,1,1,5280,39594,90260,90260,39594,5280,1,1,40814,361086,
%U A156141 981104,1288040,981104,361086,40814,1,1,363884,3801180,12088796
%N A156141 Triangle T(n,k) = A008517(n,k+1)+A008517(n,n+1-k) read by rows.
%C A156141 Row sums are 2*A001147(n).
%H A156141 E. W. Weisstein, <a href="https://mathworld.wolfram.com/Second-OrderEulerianTriangle.html">Second-Order Eulerian Triangle</a>
%F A156141 T(n,0)=T(n,n) =1, n>=1.
%F A156141 T(n,k)= A008517(n,k+1)+A008517(n,n+1-k), n>1, 0<k<n.
%F A156141 T(n,k)= T(n,n-k) .
%e A156141 {2},
%e A156141 {1, 1},
%e A156141 {1, 4, 1},
%e A156141 {1, 14, 14, 1},
%e A156141 {1, 46, 116, 46, 1},
%e A156141 {1, 172, 772, 772, 172, 1},
%e A156141 {1, 834, 5160, 8800, 5160, 834, 1},
%e A156141 {1, 5280, 39594, 90260, 90260, 39594, 5280, 1},
%e A156141 {1, 40814, 361086, 981104, 1288040, 981104, 361086, 40814, 1},
%e A156141 {1, 363884, 3801180, 12088796, 18205564, 18205564, 12088796, 3801180, 363884, 1},
%e A156141 {1, 3630826, 44557888, 167513072, 283669904, 310714768, 283669904, 167513072, 44557888, 3630826, 1}
%t A156141 e[n_, 0] := 1;
%t A156141 e[n_, k_] := 0 /; k >= n;
%t A156141 e[n_, k_] := (k + 1)e[n - 1, k] + (2n - k - 1)e[n - 1, k - 1];
%t A156141 Table[Table[e[n, k] + e[n, n - k], {k, 0, n}], {n, 0, 10}];
%t A156141 Flatten[%]
%Y A156141 Cf. A008517
%K A156141 nonn,tabl
%O A156141 0,1
%A A156141 _Roger L. Bagula_, Feb 04 2009
%E A156141 Definition simplified by the Associate Editors of the OEIS Sep 17 2009