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A363400 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) * ((2 - (n mod 2)) * j + 1)^n).

%I A363400 #18 Oct 06 2023 04:32:16
%S A363400 1,3,2,7,36,25,15,88,135,64,31,2106,10000,14406,6561,63,1824,10206,
%T A363400 22528,21875,7776,127,87480,1546875,7529536,15411789,14172488,4826809,
%U A363400 255,31616,478953,2670592,7265625,10357632,7411887,2097152
%N A363400 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) * ((2 - (n mod 2)) * j + 1)^n).
%C A363400 In A363398 we give an inclusion-exclusion representation for 2^n*Euler(n), and in A363399 we give such a representation of 2^n*Euler(n, 1) = A155585(n). Here the two representations are combined into one of A000111.
%H A363400 <a href="/index/Eu#Euler">Index entries for sequences related to Euler numbers.</a>
%F A363400 T(n, k) = A363399(n, k) for 0 <= k <= n if n is odd otherwise A363398(n, k).
%F A363400 (Sum_{k=0..n} (-1)^k * T(n, k)) / h(n) = A000111(n), where h(n) = (-1)^binomial(n, 2) * 2^(n * iseven(n)), see A059222.
%F A363400 From _Detlef Meya_, Oct 04 2023: (Start)
%F A363400 T(n, k) = (2*k + 1 - k*(n mod 2))^(n - 1)*add(binomial(n + 1, j), j = 0..n - k)*(2*k + 1 - k*(n mod 2)).
%F A363400 T(n, k) = (2^(n + 1) - binomial(n + 1, n - k + 1)*hypergeom([1, -k], [n - k + 2], -1))*(2*k + 1 - k*(n mod 2))^n. (End)
%e A363400 Triangle T(n, k) starts:
%e A363400 [0]   1;
%e A363400 [1]   3,     2;
%e A363400 [2]   7,    36,      25;
%e A363400 [3]  15,    88,     135,      64;
%e A363400 [4]  31,  2106,   10000,   14406,     6561;
%e A363400 [5]  63,  1824,   10206,   22528,    21875,     7776;
%e A363400 [6] 127, 87480, 1546875, 7529536, 15411789, 14172488, 4826809;
%e A363400 [7] 255, 31616,  478953, 2670592,  7265625, 10357632, 7411887, 2097152;
%p A363400 P := (n, x) -> add(add(x^j * binomial(k, j) * ((2 - irem(n, 2)) * j + 1)^n,
%p A363400 j = 0..k) * 2^(n - k), k = 0..n): T := (n, k) -> coeff(P(n, x), x, k):
%p A363400 seq(seq(T(n, k), k = 0..n), n = 0..8);
%t A363400 From _Detlef Meya_, Oct 04 2023: (Start)
%t A363400 T[n_, k_] := (2^(n+1)-Binomial[n+1, n-k+1]*Hypergeometric2F1[1, -k, n-k+2, -1])*(2*k+1-k*Mod[n, 2])^n;
%t A363400 (* Or: *)
%t A363400 T[n_, k_] := (2*k+1-k*Mod[n, 2])^(n-1)*Sum[Binomial[n+1, j], {j, 0, n-k}]*(2*k+1-k*Mod[n, 2]);
%t A363400 Flatten[Table[T[n, k], {n, 0, 7}, {k, 0, n}]]  (* End *)
%Y A363400 Cf. A126646 (column 0), A363401 (row sums), A000111, A059222, A002436.
%Y A363400 Cf. A363398 (secant case), A363399 (tangent case).
%K A363400 nonn,tabl
%O A363400 0,2
%A A363400 _Peter Luschny_, May 31 2023