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A380056 E.g.f. (exp(x) - 1)/cos(2*x).

%I A380056 #10 Feb 18 2025 07:36:40
%S A380056 1,1,13,25,441,1261,30213,115025,3529201,16792021,629401213,
%T A380056 3593565625,159175688361,1060279600381,54189700721013,412526870321825,
%U A380056 23894940183997921,204641610743378341,13248060325188261613,126065945039257743625,9020317522757414377881,94419130586604915837901
%N A380056 E.g.f. (exp(x) - 1)/cos(2*x).
%C A380056 Conjecture: a(n) == 1 (mod 4) for n >= 1.
%C A380056 Conjecture: a(4*n) is divisible by 5 for n >= 1.
%H A380056 Paul D. Hanna, <a href="/A380056/b380056.txt">Table of n, a(n) for n = 1..400</a>
%F A380056 E.g.f. A(x) = Sum_{n>=1} a(n)*x^n/n! satisfies the following formulas.
%F A380056 (1) A(x) = (exp(x) - 1)/cos(2*x).
%F A380056 (2) A(F(x)) = x where F(x) = log(1 + x*cos(2*F(x))) equals the e.g.f. of A380555.
%F A380056 (3.a) a(2*n+1) = Sum_{k=0..n} binomial(2*n+1, 2*k+1) * A000364(n-k) * 4^(n-k) for n >= 0.
%F A380056 (3.b) a(2*n) = Sum_{k=1..n} binomial(2*n, 2*k) * A000364(n-k) * 4^(n-k) for n >= 1.
%F A380056 a(n) ~ (exp(Pi/4) - 1 + (exp(-Pi/4) - 1)*(-1)^n) * 2^(2*n + 3/2) * n^(n + 1/2) / (sqrt(Pi) * exp(n) * Pi^n). - _Vaclav Kotesovec_, Jan 29 2025
%e A380056 E.g.f.: A(x) = x + x^2/2! + 13*x^3/3! + 25*x^4/4! + 441*x^5/5! + 1261*x^6/6! + 30213*x^7/7! + 115025*x^8/8! + 3529201*x^9/9! + 16792021*x^10/10! + ...
%o A380056 (PARI) {a(n) = my(A,X = x + x*O(x^n)); n!*polcoef( (exp(X) - 1)/cos(2*X), n)}
%o A380056 for(n=1,25,print1(a(n),", "))
%Y A380056 Cf. A380053, A000364, A380555.
%K A380056 nonn
%O A380056 1,3
%A A380056 _Paul D. Hanna_, Jan 28 2025