A354520
Expansion of e.g.f. exp(x) * log(cosh(x)).
Original entry on oeis.org
0, 1, 3, 4, 0, 1, 63, 64, -1320, -1319, 49203, 49204, -2653560, -2653559, 196707423, 196707424, -19194804720, -19194804719, 2385684870723, 2385684870724, -367985503366800, -367985503366799, 68980888889771103, 68980888889771104, -15445553274667315800
Offset: 1
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my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(exp(x)*log(cosh(x)))))
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a(n) = sum(k=1, n\2, (16^k-4^k)*bernfrac(2*k)/(2*k)*binomial(n, 2*k));
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from math import comb
from sympy import bernoulli
def A354520(n): return sum((((2<<(m:=k<<1))-2)*bernoulli(m)<>1)+1)) # Chai Wah Wu, Apr 15 2023
A215518
2n-th derivative of cosh(x)^cosh(x) at x=0.
Original entry on oeis.org
1, 1, 7, 76, 1597, 41776, 1673167, 74527636, 4832747017, 305644428256, 30618856073947, 2276081971574236, 390042814538656957, 20435946140834126176, 10544180964356207226247, -604793906292405974180324, 688972694565220644332739217, -181007844268190205159712489664
Offset: 0
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a:= n-> coeff(series(cosh(x)^cosh(x), x, 2*n+1), x, 2*n)*(2*n)!:
seq(a(n), n=0..21); # Alois P. Heinz, Aug 15 2012, revised, Apr 14 2023
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f[x_] := Cosh[x]^Cosh[x]; Table[Derivative[2*n][f][0],{n,0,30}]
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a354520(n) = sum(k=1, n\2, (16^k-4^k)*bernfrac(2*k)/(2*k)*binomial(n, 2*k));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, a354520(2*j)*binomial(2*i-1, 2*j-1)*v[i-j+1])); v; \\ Seiichi Manyama, Aug 17 2022
A354517
Expansion of e.g.f. cos(x)^exp(x).
Original entry on oeis.org
1, 0, -1, -3, -5, 10, 134, 742, 2325, -2820, -118756, -1138368, -7132025, -20945990, 196411214, 4438271692, 50498101545, 400644382200, 1571151012344, -16415635331328, -500300343321365, -7486919544207050, -81415563206142166, -563533196469890228
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(cos(x)^exp(x)))
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a354519(n) = sum(k=1, n\2, ((-4)^k-(-16)^k)*bernfrac(2*k)/(2*k)*binomial(n, 2*k));
a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-sum(j=1, i, a354519(j)*binomial(i-1, j-1)*v[i-j+1])); v;
Showing 1-3 of 3 results.