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A378066 Array read by ascending antidiagonals: A(n, k) = (-2*n)^k * Euler(k, (n - 1)/(2*n)) for n >= 1 and A(0, k) = 1.

%I A378066 #11 Nov 18 2024 13:01:19
%S A378066 1,1,1,1,1,1,1,1,0,1,1,1,-3,-2,1,1,1,-8,-11,0,1,1,1,-15,-26,57,16,1,1,
%T A378066 1,-24,-47,352,361,0,1,1,1,-35,-74,1185,1936,-2763,-272,1,1,1,-48,
%U A378066 -107,2976,6241,-38528,-24611,0,1
%N A378066 Array read by ascending antidiagonals: A(n, k) = (-2*n)^k * Euler(k, (n - 1)/(2*n)) for n >= 1 and A(0, k) = 1.
%C A378066 This is the counterpart of A377666, where A(1, n) are the secant numbers A122045(n). Here A(1, n) are the tangent numbers A155585(n).
%F A378066 A(n, k) = k! * [x^k] exp(x)/cosh(n*x).
%F A378066 A(n, k) = Sum_{j = 0..k} binomial(k, j) * Euler(j, 1/2) *(-2*n)^j.
%e A378066 Array starts:
%e A378066   [0]  1, 1,   1,    1,     1,     1,        1, ...  A000012
%e A378066   [1]  1, 1,   0,   -2,     0,    16,        0, ...  A155585
%e A378066   [2]  1, 1,  -3,  -11,    57,   361,    -2763, ...  A188458
%e A378066   [3]  1, 1,  -8,  -26,   352,  1936,   -38528, ...  A000810
%e A378066   [4]  1, 1, -15,  -47,  1185,  6241,  -230895, ...  A000813
%e A378066   [5]  1, 1, -24,  -74,  2976, 15376,  -906624, ...  A378065
%e A378066   [6]  1, 1, -35, -107,  6265, 32041, -2749355, ...
%e A378066   [7]  1, 1, -48, -146, 11712, 59536, -6997248, ...
%p A378066 A := (n, k) -> ifelse(n = 0, 1, (-2*n)^k * euler(k, (n - 1) / (2*n))):
%p A378066 for n from 0 to 7 do seq(A(n, k), k = 0..9) od; # row by row
%p A378066 # Alternative:
%p A378066 A := proc(n, k) local j; add(binomial(k, j)*euler(j, 1/2)*(-2*n)^j, j = 0..k) end: seq(seq(A(n - k, k), k = 0..n), n = 0..10);
%p A378066 # Using generating functions:
%p A378066 egf := n -> exp(x)/cosh(n*x): ser := n -> series(egf(n), x, 14):
%p A378066 row := n -> local k; seq(k!*coeff(ser(n), x, k), k = 0..7):
%p A378066 seq(lprint(row(n)), n = 0..7);
%Y A378066 Rows: A000012, A155585, A188458, A000810, A000813, A378065.
%Y A378066 Columns: A005563 (k=2), A080663 (k=3), A378064 (k=4).
%Y A378066 Cf. A378063 (main diagonal), A377666 (secant), A081658 (column generating polynomials).
%K A378066 sign,tabl
%O A378066 0,13
%A A378066 _Peter Luschny_, Nov 15 2024