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Expansion of e.g.f. exp(x) * log(cosh(x)).
5

%I #28 Apr 15 2023 15:53:56

%S 0,1,3,4,0,1,63,64,-1320,-1319,49203,49204,-2653560,-2653559,

%T 196707423,196707424,-19194804720,-19194804719,2385684870723,

%U 2385684870724,-367985503366800,-367985503366799,68980888889771103,68980888889771104,-15445553274667315800

%N Expansion of e.g.f. exp(x) * log(cosh(x)).

%H Seiichi Manyama, <a href="/A354520/b354520.txt">Table of n, a(n) for n = 1..486</a>

%F a(n) = Sum_{k=1..floor(n/2)} (-1)^(k+1) * A000182(k) * binomial(n,2*k).

%F a(2*n) = a(2*n-1) + 1.

%o (PARI) my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(exp(x)*log(cosh(x)))))

%o (PARI) a(n) = sum(k=1, n\2, (16^k-4^k)*bernfrac(2*k)/(2*k)*binomial(n, 2*k));

%o (Python)

%o from math import comb

%o from sympy import bernoulli

%o def A354520(n): return sum((((2<<(m:=k<<1))-2)*bernoulli(m)<<m-2)//k*comb(n,k<<1) for k in range(1,(n>>1)+1)) # _Chai Wah Wu_, Apr 15 2023

%Y Cf. A000182, A354518, A354519.

%K sign

%O 1,3

%A _Seiichi Manyama_, Aug 16 2022