A352644
Expansion of e.g.f. exp(2 * x * cosh(x)).
Original entry on oeis.org
1, 2, 4, 14, 64, 282, 1504, 9382, 59392, 417842, 3234304, 25854270, 222423040, 2039652682, 19398688768, 195195414998, 2058666508288, 22509119309922, 257507944431616, 3058883865391726, 37588692110934016, 479523015556635002, 6316374397705781248
Offset: 0
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With[{m = 22}, Range[0, m]! * CoefficientList[Series[Exp[2*x*Cosh[x]], {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(2*x*cosh(x))))
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a(n) = if(n==0, 1, 2*sum(k=0, (n-1)\2, (2*k+1)*binomial(n-1, 2*k)*a(n-2*k-1)));
A352645
Expansion of e.g.f. exp(3 * x * cosh(x)).
Original entry on oeis.org
1, 3, 9, 36, 189, 1068, 6669, 47568, 363897, 2976048, 26422929, 248498880, 2464625205, 25922981568, 285884886933, 3296418423552, 39795656126193, 499998631895808, 6526830528863001, 88493793850020864, 1241919513823360941, 18020032474632956928
Offset: 0
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With[{m = 21}, Range[0, m]! * CoefficientList[Series[Exp[3*x*Cosh[x]], {x, 0, m}], x]] (* Amiram Eldar, Mar 26 2022 *)
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my(N=40, x='x+O('x^N)); Vec(serlaplace(exp(3*x*cosh(x))))
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a(n) = if(n==0, 1, 3*sum(k=0, (n-1)\2, (2*k+1)*binomial(n-1, 2*k)*a(n-2*k-1)));
A381341
Expansion of e.g.f. exp( x * cosh(sqrt(2)*x) ).
Original entry on oeis.org
1, 1, 1, 7, 25, 81, 601, 3207, 18705, 156385, 1087441, 8962823, 84001897, 732712241, 7487525865, 78537490951, 831722893217, 9804469109953, 115549730623009, 1431784628480007, 18795444460125241, 248964703826005777, 3487888859183694329, 50283005924345951111
Offset: 0
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a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, 2^((n-k)/2)*a185951(n, k));
A381342
Expansion of e.g.f. exp( x * cos(sqrt(2)*x) ).
Original entry on oeis.org
1, 1, 1, -5, -23, -39, 361, 2675, 3697, -90575, -741839, 52779, 48483865, 358510985, -1225182503, -43006420829, -239523048095, 2745896185953, 54532102774753, 144304368441179, -6547928921714999, -88336890555248327, 199686588300036553, 18186115601328322515
Offset: 0
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a185951(n, k) = binomial(n, k)/2^k*sum(j=0, k, (2*j-k)^(n-k)*binomial(k, j));
a(n) = sum(k=0, n, (-2)^((n-k)/2)*a185951(n, k));
A307996
Expansion of e.g.f. exp(1 - exp(x)*(1 - 2*x)).
Original entry on oeis.org
1, 1, 4, 15, 73, 410, 2591, 18165, 139266, 1155509, 10293729, 97815520, 986113613, 10499247005, 117603042220, 1381191356979, 16958788930317, 217132031279842, 2892337840164051, 40002168264724193, 573363461815952802, 8502905138072937073, 130268705062115090965, 2058969680487762098496
Offset: 0
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nmax = 23; CoefficientList[Series[Exp[1 - Exp[x] (1 - 2 x)], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = Sum[(2 k - 1) Binomial[n - 1, k - 1] a[n - k], {k, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 23}]
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