A349103
Expansion of e.g.f. exp( (sin(x) + x*cos(x))/2 ).
Original entry on oeis.org
1, 1, 1, -1, -7, -16, 19, 270, 809, -1832, -26939, -81680, 415817, 5085192, 11943199, -154023064, -1510744111, -1192784896, 80798373289, 615743987520, -1348945040647, -55570193758592, -296824726927253, 2567750854163200, 47111874571146041
Offset: 0
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With[{m = 24}, Range[0, m]! * CoefficientList[Series[Exp[(Sin[x] + x*Cos[x])/2], {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp((sin(x)+x*cos(x))/2)))
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a(n) = if(n==0, 1, sum(k=0, (n-1)\2, (-1)^k*(k+1)*binomial(n-1, 2*k)*a(n-2*k-1)));
A349105
Expansion of e.g.f. 1/(1 - (sinh(x) + x*cosh(x))/2 ).
Original entry on oeis.org
1, 1, 2, 8, 40, 243, 1796, 15502, 152608, 1690613, 20814208, 281859540, 4163795648, 66636761575, 1148477490304, 21207704998010, 417728195909632, 8742243282090153, 193720478508563456, 4531158728871170080, 111562803180301643776, 2884156736234559267611
Offset: 0
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With[{m = 21}, Range[0, m]! * CoefficientList[Series[1/(1 - (Sinh[x] + x*Cosh[x])/2), {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)
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my(N=40, x='x+O('x^N)); Vec(serlaplace(1/(1-(sinh(x)+x*cosh(x))/2)))
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a(n) = if(n==0, 1, sum(k=0, (n-1)\2, (k+1)*binomial(n, 2*k+1)*a(n-2*k-1)));
A352254
Expansion of e.g.f. exp( x * sinh(x) / 2 ) (even powers only).
Original entry on oeis.org
1, 1, 5, 48, 753, 16880, 507579, 19509042, 927229553, 53126200872, 3597373129635, 283321938437318, 25614466939850169, 2629191169850594388, 303549146372282854883, 39103024746814973908890, 5581172267077778765676129, 877211696663645448333041072, 151002471269513108372760683523
Offset: 0
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nmax = 36; Take[CoefficientList[Series[Exp[x Sinh[x]/2], {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
a[0] = 1; a[n_] := a[n] = Sum[Binomial[2 n - 1, 2 k - 1] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
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my(x='x+O('x^40), v=Vec(serlaplace(exp(x*sinh(x)/2)))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022
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