A335629
Expansion of e.g.f. Product_{k>0} (1+sin(x)^k).
Original entry on oeis.org
1, 1, 2, 11, 40, 241, 1952, 13691, 96640, 1124161, 13770752, 119165771, 1373271040, 21966388081, 297353879552, 4935774720251, 71240808202240, 589721910720001, 20308058536804352, 594276723452229131, 4142005120030474240, 73558191271194967921, 5026768573876078641152, 36395755978909448444411, -115943295497314720808960
Offset: 0
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nmax = 25; CoefficientList[Series[Product[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(eta(sin(x)^2)/eta(sin(x))))
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N=40; x='x+O('x^N); Vec(serlaplace(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, (-sin(x))^k/(k*(sin(x)^k-1))))))
A335627
Expansion of e.g.f. Product_{k>0} 1/(1-tan(x)^k).
Original entry on oeis.org
1, 1, 4, 20, 152, 1216, 13264, 145760, 2031872, 28617856, 480749824, 8243878400, 162085486592, 3262756228096, 73483961257984, 1695754607421440, 42992308610957312, 1118097332524711936, 31487163119164063744, 910421423509984378880, 28187970433553669292032, 896242635855128514789376
Offset: 0
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nmax = 25; CoefficientList[Series[Product[1/(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(1/eta(tan(x))))
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N=40; x='x+O('x^N); Vec(serlaplace(1/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, sigma(k)*tan(x)^k/k))))
A335635
Expansion of e.g.f. Product_{k>0} 1/(1 - sin(x)^k / k).
Original entry on oeis.org
1, 1, 3, 10, 44, 215, 1252, 7992, 56024, 438341, 3672328, 32587366, 318586880, 3325053147, 35115462592, 407034567076, 5198294627456, 63965057355305, 824995119961984, 12611299833296898, 184189806819806720, 2590874864719588031, 44912343151409875456, 728583107189913021328, 11458864344772729650176
Offset: 0
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max = 24; Range[0, max]! * CoefficientList[Series[Product[1/(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(1/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, sin(x)^(i*j)/(i*j^i))))))
A346841
E.g.f.: Product_{k>=1} 1 / (1 - x^k)^sin(x).
Original entry on oeis.org
1, 0, 2, 9, 40, 360, 2480, 28833, 266936, 3562920, 45634258, 659631225, 10231705196, 176661237948, 3080315922294, 59430009554685, 1217593208993232, 25766943601055184, 583245289316927058, 13861911731632256457, 343615639889119016556, 8925102256331257339140, 242399591002192962709230
Offset: 0
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N=40; x='x+O('x^N); Vec(serlaplace(1/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)*x^k/k))))
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N=40; x='x+O('x^N); Vec(serlaplace(exp(sin(x)*sum(k=1, N, x^k/(k*(1-x^k))))))
A335642
Expansion of e.g.f. Product_{k>0} 1/(1 - sin(x)^k / k!).
Original entry on oeis.org
1, 1, 3, 9, 35, 147, 710, 3780, 21391, 136063, 932190, 6887232, 55902274, 497726270, 4711586833, 47692742905, 528539419087, 6093676850975, 73010887114406, 943925266298096, 12740929019736310, 175037826035276730, 2561985529052306447, 39817440376814520907, 622315443336146270858
Offset: 0
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max = 24; Range[0, max]! * CoefficientList[Series[Product[1/(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(1/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, sin(x)^(i*j)/(i*j!^i))))))
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