A354518
Expansion of e.g.f. cosh(x)^exp(x).
Original entry on oeis.org
1, 0, 1, 3, 7, 30, 166, 798, 4117, 27660, 196756, 1328448, 9866407, 86205210, 759842266, 6460661028, 60841732777, 651349676280, 6795873687496, 67981177154688, 770224145659627, 9854500496860470, 116983085896035646, 1301594922821009028, 17440543467561038557
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(cosh(x)^exp(x)))
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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(j)*binomial(i-1, j-1)*v[i-j+1])); v;
A215515
2n-th derivative of cos(x)^cos(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
Offset: 0
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a:= n-> coeff(series(cos(x)^cos(x), x, 2*n+1), x, 2*n)*(2*n)!:
seq(a(n), n=0..21); # Alois P. Heinz, Apr 17 2023
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f[x_] := Cos[x]^Cos[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, (-1)^j*a354520(2*j)*binomial(2*i-1, 2*j-1)*v[i-j+1])); v; \\ Seiichi Manyama, Aug 17 2022
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
A354519
Expansion of e.g.f. exp(x) * log(sec(x)).
Original entry on oeis.org
0, 1, 3, 8, 20, 61, 203, 888, 4080, 24001, 140283, 1028048, 7248020, 63374221, 522164243, 5299033488, 49924707840, 576514338721, 6110861416083, 79100066353208, 931434877343540, 13355627237749501, 172948115797623803, 2720827878727067208, 38424408320191299120
Offset: 1
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my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(exp(x)*log(1/cos(x)))))
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a(n) = sum(k=1, n\2, ((-4)^k-(-16)^k)*bernfrac(2*k)/(2*k)*binomial(n, 2*k));
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from math import comb
from sympy import bernoulli
def A354519(n): return sum(abs(((2-(2<<(m:=k<<1)))*bernoulli(m)<>1)+1)) # Chai Wah Wu, Apr 15 2023
Showing 1-4 of 4 results.
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