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A102365 Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...] where DELTA is the operator defined in A084938.

%I A102365 #25 Jan 22 2020 21:07:13
%S A102365 1,1,0,1,1,0,1,5,1,0,1,18,15,1,0,1,58,129,37,1,0,1,179,877,646,83,1,0,
%T A102365 1,543,5280,8030,2685,177,1,0,1,1636,29658,82610,56285,10002,367,1,0,
%U A102365 1,4916,159742,756218,919615,335162,34777,749,1,0
%N A102365 Triangle T(n,k), 0 <= k <= n, read by rows: given by [ 1, 0, 3, 0, 5, 0, 7, 0, 9, 0, ...] DELTA [ 0, 1, 0, 2, 0, 3, 0, 4, 0, 5, 0, ...] where DELTA is the operator defined in A084938.
%C A102365 Generalized Eulerian numbers A008292.
%C A102365 Reversal of A211399. - _Philippe Deléham_, Feb 12 2013
%H A102365 G. C. Greubel, <a href="/A102365/b102365.txt">Table of n, a(n) for the first 50 rows, flattened</a>
%F A102365 T(n, k) = (n-k)*T(n-1, k-1) + (2*k+1)*T(n-1, k) with T(0, 0) = 1, T(0, k) = 0 if k > 0, T(n, k) = 0 if k < 0.
%F A102365 Sum_{k>=0} T(n, k)*2^k = A001147(n).
%F A102365 Sum_{k>=0} T(n, k) = A014307(n). - _Philippe Deléham_, Mar 19 2005
%e A102365 Triangle begins:
%e A102365   1;
%e A102365   1,  0;
%e A102365   1,  1,   0;
%e A102365   1,  5,   1,  0;
%e A102365   1, 18,  15,  1, 0;
%e A102365   1, 58, 129, 37, 1, 0; ...
%t A102365 T[0, 0] := 1;  T[n_, -1] := 0;  T[n_, n_] := 0; T[n_, k_] := T[n, k] = (n - k)*T[n - 1, k - 1] + (2*k + 1)*T[n - 1, k]; Join[{1}, Table[If[k < 0, 0, If[k >= n, 0, T[n, k]]], {n, 1, 5}, {k, 0, n}] // Flatten] (* _G. C. Greubel_, Jun 30 2017 *)
%Y A102365 Diagonals: A000007, A000012, A050488, A142965, A142966.
%Y A102365 Columns: A000012, A000340, A156922, A156923, A156924.
%K A102365 nonn,easy,tabl
%O A102365 0,8
%A A102365 _Philippe Deléham_, Feb 22 2005