A335626
Expansion of e.g.f. Product_{k>0} 1/(1-sin(x)^k).
Original entry on oeis.org
1, 1, 4, 17, 104, 661, 5584, 47837, 483584, 5332681, 63940864, 802442057, 11548580864, 170258934301, 2602357970944, 44379608478677, 800966933970944, 14221966162901521, 277738909303373824, 5823354583392253697, 121050262784565837824, 2668717158207399650341, 62376912442894992277504
Offset: 0
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nmax = 25; CoefficientList[Series[Product[1/(1 - Sin[x]^k), {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 03 2020 *)
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N=40; x='x+O('x^N); Vec(serlaplace(1/eta(sin(x))))
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N=40; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, 1-sin(x)^k)))
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N=40; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, sigma(k)*sin(x)^k/k))))
A335630
Expansion of e.g.f. Product_{k>0} (1+tan(x)^k).
Original entry on oeis.org
1, 1, 2, 14, 64, 616, 5072, 58064, 669184, 9417856, 137019392, 2294104064, 40350383104, 778782954496, 16050760435712, 352024447115264, 8269739647565824, 204097141026881536, 5360540853755052032, 147190808628196081664, 4270498402940171321344, 129024432217526266494976
Offset: 0
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nmax = 25; CoefficientList[Series[Product[1 + Tan[x]^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 03 2020 *)
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N=40; x='x+O('x^N); Vec(serlaplace(eta(tan(x)^2)/eta(tan(x))))
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N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, 1+tan(x)^k)))
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N=40; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, (-tan(x))^k/(k*(tan(x)^k-1))))))
A335637
Expansion of e.g.f. Product_{k>0} (1 + sin(x)^k / k).
Original entry on oeis.org
1, 1, 1, 4, 10, 25, 210, 978, 2336, 25265, 361424, 1557752, -1098528, 140915385, 2093367328, 10484632486, 133131785728, -1343478380255, -8738565516288, 1790935681747980, 3245598828836864, -592809746388403495, 6832010190766985216, 179327221659613996634, -5310378915096702812160
Offset: 0
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max = 24; Range[0, max]! * CoefficientList[Series[Product[1 + Sin[x]^k/k, {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Oct 03 2020 *)
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N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, 1+sin(x)^k/k)))
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N=40; x='x+O('x^N); Vec(serlaplace(exp(sum(i=1, N, sum(j=1, N\i, (-1)^(i+1)*sin(x)^(i*j)/(i*j^i))))))
A335644
Expansion of e.g.f. Product_{k>0} (1 + sin(x)^k / k!).
Original entry on oeis.org
1, 1, 1, 3, 1, -23, -2, -28, -2435, 253, 118966, 158400, -5277106, -6453094, 377003877, 150562341, -38919169331, -49489639843, 4097920244054, 15989402021656, -397866849121614, -3949517739363706, 34992745696351023, 937723673130987379, -2417716098650478930, -223227071403982903362
Offset: 0
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max = 25; Range[0, max]! * CoefficientList[Series[Product[1 + Sin[x]^k/k!, {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Oct 04 2020 *)
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N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, 1+sin(x)^k/k!)))
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N=40; x='x+O('x^N); Vec(serlaplace(exp(sum(i=1, N, sum(j=1, N\i, (-1)^(i+1)*sin(x)^(i*j)/(i*j!^i))))))
A347817
E.g.f.: Product_{k>=1} (1 + x^k)^sin(x).
Original entry on oeis.org
1, 0, 2, 3, 40, 80, 1760, 8211, 139256, 763272, 19466578, 147696835, 3372858476, 33370016316, 872184749046, 10340382875655, 289042962136272, 3884706041971728, 118640349946950738, 1911641854423398435, 59577007012206421356, 1086774235381609797540, 37138839666110194130670
Offset: 0
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N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, (1+x^k)^sin(x))))
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N=40; x='x+O('x^N); Vec(serlaplace(exp(sin(x)*sum(k=1, N, sigma(k>>valuation(k, 2))*x^k/k))))
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