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A363394 Triangle read by rows. T(n, k) = A081658(n, k) + A363393(n, k) for k > 0 and T(n, 0) = 1.

%I A363394 #20 Nov 17 2024 08:30:13
%S A363394 1,1,1,1,2,-1,1,3,-3,-2,1,4,-6,-8,5,1,5,-10,-20,25,16,1,6,-15,-40,75,
%T A363394 96,-61,1,7,-21,-70,175,336,-427,-272,1,8,-28,-112,350,896,-1708,
%U A363394 -2176,1385,1,9,-36,-168,630,2016,-5124,-9792,12465,7936
%N A363394 Triangle read by rows. T(n, k) = A081658(n, k) + A363393(n, k) for k > 0 and T(n, 0) = 1.
%H A363394 Richard P. Stanley, <a href="https://doi.org/10.48550/arXiv.0912.4240">A survey of alternating permutations</a>, arXiv:0912.4240 [math.CO], 2009.
%H A363394 <a href="/index/Eu#Euler">Index entries for sequences related to Euler numbers.</a>
%F A363394 |T(n, k)| = (-1)^(n - k) * n! * [x^(n - k)][y^n] (sec(y) + tan(y)) / exp(x*y).
%F A363394 T(n, k) = [x^(n - k)] -2^(k-(0^k))*(Euler(k, 0) + Euler(k, 1/2)) / (x-1)^(k + 1).
%F A363394 For a recursion see the Python program.
%F A363394 T(n, k) = [x^n] ((-1) + Sum_{j=0..n} binomial(n, j)*(Euler(j, 1) + Euler(j, 1/2))*(2*x)^j). - _Peter Luschny_, Nov 17 2024
%e A363394 The triangle T(n, k) begins:
%e A363394   [0] 1;
%e A363394   [1] 1, 1;
%e A363394   [2] 1, 2,  -1;
%e A363394   [3] 1, 3,  -3,   -2;
%e A363394   [4] 1, 4,  -6,   -8,   5;
%e A363394   [5] 1, 5, -10,  -20,  25,   16;
%e A363394   [6] 1, 6, -15,  -40,  75,   96,   -61;
%e A363394   [7] 1, 7, -21,  -70, 175,  336,  -427,  -272;
%e A363394   [8] 1, 8, -28, -112, 350,  896, -1708, -2176,  1385;
%e A363394   [9] 1, 9, -36, -168, 630, 2016, -5124, -9792, 12465, 7936;
%p A363394 # Variant, computes abs(T(n, k)):
%p A363394 P := n -> n!*coeff(series((sec(y) + tan(y))/exp(x*y), y, 24), y, n):
%p A363394 seq(print(seq((-1)^(n - k)*coeff(P(n), x, n - k), k = 0..n)), n = 0..9);
%o A363394 (Python)
%o A363394 from functools import cache
%o A363394 @cache
%o A363394 def T(n: int, k: int) -> int:
%o A363394     if k == 0: return 1
%o A363394     if k == n:
%o A363394         p = k % 2
%o A363394         return p - sum(T(n, j) for j in range(p, n - 1, 2))
%o A363394     return (T(n - 1, k) * n) // (n - k)
%o A363394 for n in range(10): print([T(n, k) for k in range(n + 1)])
%Y A363394 Variants (row reversed): A109449, A247453.
%Y A363394 Cf. A081658 (signed secant part), A363393 (signed tangent part), A000111 (main diagonal), A122045, A155585 (aerated main diagonal), A000667, A062162 (row sums of signless variant).
%K A363394 sign,tabl
%O A363394 0,5
%A A363394 _Peter Luschny_, Jun 06 2023