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A363398 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*j + 1)^n), (secant case).

%I A363398 #24 Mar 27 2025 05:43:20
%S A363398 1,3,3,7,36,25,15,297,625,343,31,2106,10000,14406,6561,63,13851,
%T A363398 131250,369754,413343,161051,127,87480,1546875,7529536,15411789,
%U A363398 14172488,4826809,255,540189,17109375,134237509,444816117,721025327,564736653,170859375
%N A363398 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*j + 1)^n), (secant case).
%C A363398 Here we give an inclusion-exclusion representation of 2^n*Euler(n) (see A122045 and A002436), in A363399 we give such a representation for 2^n*Euler(n, 1) = A155585(n), and in A363400 one for the combined sequences.
%H A363398 <a href="/index/Eu#Euler">Index entries for sequences related to Euler numbers.</a>
%F A363398 Sum_{k=0..n} (-1)^k*T(n, k) = 2^n*Euler(n) = 4^n*Euler(n, 1/2).
%F A363398 (Sum_{k=0..n} (-1)^k*T(n, k)) / 2^n = Euler(n) = 2^n*Euler(n, 1/2) = A122045(n).
%F A363398 Sum_{k=0..2*n} (-1)^k*T(2*n, k) = 4^n*Euler(2*n) = 16^n*Euler(2*n, 1/2) = (-1)^n*A002436(n).
%F A363398 From _Detlef Meya_, Oct 04 2023: (Start)
%F A363398 T(n, k) = (2*k + 1)^n * binomial(n+1, k+1) * hypergeom([1, k-n], [k+2], -1).
%F A363398 T(n, k) = (2*k + 1)^n * (2^(n + 1) - Sum_{j=0..k} binomial(n+1, j)). (End)
%e A363398 The triangle T(n, k) starts:
%e A363398   [0]   1;
%e A363398   [1]   3,      3;
%e A363398   [2]   7,     36,       25;
%e A363398   [3]  15,    297,      625,       343;
%e A363398   [4]  31,   2106,    10000,     14406,      6561;
%e A363398   [5]  63,  13851,   131250,    369754,    413343,    161051;
%e A363398   [6] 127,  87480,  1546875,   7529536,  15411789,  14172488,   4826809;
%e A363398   [7] 255, 540189, 17109375, 134237509, 444816117, 721025327, 564736653, 170859375;
%p A363398 P := (n, x) -> add(add(x^j*binomial(k, j)*(2*j + 1)^n, j=0..k)*2^(n-k), k=0..n):
%p A363398 T := (n, k) -> coeff(P(n, x), x, k): seq(seq(T(n, k), k = 0..n), n = 0..7);
%t A363398 (* From _Detlef Meya_, Oct 04 2023: (Start) *)
%t A363398 T[n_, k_] := (2*k+1)^n*(2^(n+1) - Sum[Binomial[n+1, j], {j,0,k}]);
%t A363398 (* Or: *)
%t A363398 T[n_, k_] := (2*k+1)^n*Binomial[n+1, k+1]*Hypergeometric2F1[1, k-n, k+2, -1];
%t A363398 Flatten[Table[T[n, k], {n, 0, 7}, {k, 0, n}]]  (* End *)
%Y A363398 Cf. A122045 (alternating row sums), A363396 (row sums), A126646 (column 0), A085527 (main diagonal), A141475 (central terms).
%Y A363398 Cf. A363399 (tangent case), A363400 (combined case).
%Y A363398 Cf. A122045, A002436.
%K A363398 nonn,tabl
%O A363398 0,2
%A A363398 _Peter Luschny_, May 31 2023