A009233
Expansion of e.g.f. exp(sinh(x)*x) (even powers only).
Original entry on oeis.org
1, 2, 16, 246, 5944, 202330, 9099564, 517447126, 36048776656, 3003924569778, 293835907664980, 33232296062419630, 4291773869167401720, 626311538509296801226, 102365694283336181089084, 18595053487766135171539590, 3729223211361742071603266464
Offset: 0
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With[{nn=30},Take[CoefficientList[Series[Exp[Sinh[x]*x],{x,0,nn}],x] Range[0,nn]!,{1,-1,2}]] (* Harvey P. Dale, Jul 31 2020 *)
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my(x='x+O('x^40), v=Vec(serlaplace(exp(sinh(x)*x)))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022
A296788
Expansion of e.g.f. exp(x*arcsinh(x)) (even powers only).
Original entry on oeis.org
1, 2, 8, 54, 104, 18810, -1648428, 247726374, -49445941200, 12841169289714, -4206667789245780, 1697448414191239710, -827415782970517712376, 479396168140498731959850, -325673237888367403728512700, 256401822876859593450127851030, -231597610351491427264049084814240
Offset: 0
exp(x*arcsinh(x)) = 1 + 2*x^2/2! + 8*x^4/4! + 54*x^6/6! + 104*x^8/8! + ...
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nmax = 16; Table[(CoefficientList[Series[Exp[x ArcSinh[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
nmax = 16; Table[(CoefficientList[Series[(x + Sqrt[1 + x^2])^x, {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
A352250
Expansion of e.g.f. 1 / (1 - x * sin(x)) (even powers only).
Original entry on oeis.org
1, 2, 20, 486, 21944, 1591210, 169207092, 24808395262, 4796420822384, 1182349445882706, 361939981107422060, 134705596642758848806, 59900689507397744253096, 31365504832631796986962426, 19102102945852191813235300004, 13387748268024668296590660222030
Offset: 0
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nmax = 30; Take[CoefficientList[Series[1/(1 - x Sin[x]), {x, 0, nmax}], x] Range[0, nmax]!, {1, -1, 2}]
a[0] = 1; a[n_] := a[n] = 2 Sum[(-1)^(k + 1) Binomial[2 n, 2 k] k a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 15}]
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my(x='x+O('x^40), v=Vec(serlaplace(1 /(1-x*sin(x))))); vector(#v\2, k, v[2*k-1]) \\ Michel Marcus, Mar 10 2022
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;
Showing 1-5 of 5 results.