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A363399 Triangle read by rows. T(n, k) = [x^k] P(n, x), where P(n, x) = Sum_{k=0..n} 2^(n - k) * Sum_{j=0..k} (x^j * binomial(k, j) * (j + 1)^n), (tangent case).

%I A363399 #20 Oct 06 2023 04:22:49
%S A363399 1,3,2,7,16,9,15,88,135,64,31,416,1296,1536,625,63,1824,10206,22528,
%T A363399 21875,7776,127,7680,72171,262144,453125,373248,117649,255,31616,
%U A363399 478953,2670592,7265625,10357632,7411887,2097152,511,128512,3057426,25034752,100000000,218350080,265180846,167772160,43046721
%N A363399 Triangle read by rows. T(n, k) = [x^k] P(n, x), where P(n, x) = Sum_{k=0..n} 2^(n - k) * Sum_{j=0..k} (x^j * binomial(k, j) * (j + 1)^n), (tangent case).
%C A363399 Here we give an inclusion-exclusion representation of 2^n*Euler(n, 1) = A155585(n), in A363398 we give such a representation for 2^n*Euler(n), and in A363400 one for the combined sequences.
%H A363399 <a href="/index/Eu#Euler">Index entries for sequences related to Euler numbers.</a>
%F A363399 Sum_{k=0..n} (-1)^k * T(n, k) = 2^n*Euler(n, 1) = (-2)^n*Euler(n, 0) = A155585(n).
%F A363399 From  _Detlef Meya_, Oct 04 2023: (Start)
%F A363399 T(n, k) = (k + 1)^n*binomial(n + 1, k + 1)*hypergeom([1, k - n], [k + 2], -1).
%F A363399 T(n, k) = (k + 1)^n * (2^(n + 1) - add(binomial(n + 1, j), j=0..k)). (End)
%e A363399 The triangle T(n, k) begins:
%e A363399   [0]   1;
%e A363399   [1]   3,     2;
%e A363399   [2]   7,    16,      9;
%e A363399   [3]  15,    88,    135,      64;
%e A363399   [4]  31,   416,   1296,    1536,     625;
%e A363399   [5]  63,  1824,  10206,   22528,   21875,     7776;
%e A363399   [6] 127,  7680,  72171,  262144,  453125,   373248,  117649;
%e A363399   [7] 255, 31616, 478953, 2670592, 7265625, 10357632, 7411887, 2097152;
%p A363399 P := (n, x) -> add(add(x^j*binomial(k, j)*(j + 1)^n, j=0..k)*2^(n - k), k = 0..n):
%p A363399 T := (n, k) -> coeff(P(n, x), x, k): seq(seq(T(n, k), k = 0..n), n = 0..8);
%t A363399 (* From  _Detlef Meya_, Oct 04 2023: (Start) *)
%t A363399 T[n_, k_] := (k+1)^n*(2^(n+1)-Sum[Binomial[n+1, j], {j, 0, k}]);
%t A363399 (* Or *)
%t A363399 T[n_, k_] := (k+1)^n*Binomial[n+1, k+1]*Hypergeometric2F1[1, k-n, k+2, -1];
%t A363399 Flatten[Table[T[n, k], {n, 0, 7}, {k, 0, n}]]  (* End *)
%Y A363399 Cf. A155585 (alternating row sums), A363397 (row sums), A126646 (column 0), A000169 (main diagonal), A163395 (central terms), A084623.
%Y A363399 Cf. A363398 (secant case), A363400 (combined case).
%K A363399 nonn,tabl
%O A363399 0,2
%A A363399 _Peter Luschny_, May 31 2023