A354545
Expansion of e.g.f. exp(x)^( cos(x) + sin(x) ).
Original entry on oeis.org
1, 1, 3, 4, 9, -24, -143, -902, -1631, 5176, 109841, 664302, 1479841, -16079764, -240229975, -1395162974, 126628545, 101950486736, 1118811398113, 4468008939542, -46600859353919, -1019505781080044, -7952038289388071, 10041106628453162
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)^(cos(x)+sin(x))))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (-1)^((j-1)\2)*j*binomial(i-1, j-1)*v[i-j+1])); v;
A354546
Expansion of e.g.f. exp(x)^( cos(x) - sin(x) ).
Original entry on oeis.org
1, 1, -1, -8, -7, 96, 385, -1210, -14943, -5912, 593361, 2409298, -22935647, -236575468, 590041257, 20313729886, 40488350401, -1659176093392, -11796304552991, 120680593857514, 1966312603184321, -4949789957167124, -288454178376442407, -849587090710029098
Offset: 0
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my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x)^(cos(x)-sin(x))))
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a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=sum(j=1, i, (-1)^(j\2)*j*binomial(i-1, j-1)*v[i-j+1])); v;
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));
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