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Expansion of e.g.f. Product_{k>0} 1/(1-sin(x)^k).
+10
6
1, 1, 4, 17, 104, 661, 5584, 47837, 483584, 5332681, 63940864, 802442057, 11548580864, 170258934301, 2602357970944, 44379608478677, 800966933970944, 14221966162901521, 277738909303373824, 5823354583392253697, 121050262784565837824, 2668717158207399650341, 62376912442894992277504
FORMULA
E.g.f.: exp( Sum_{k>0} sigma(k)*sin(x)^k/k ).
MATHEMATICA
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 *)
PROG
(PARI) N=40; x='x+O('x^N); Vec(serlaplace(1/eta(sin(x))))
(PARI) N=40; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, 1-sin(x)^k)))
(PARI) N=40; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, sigma(k)*sin(x)^k/k))))
Expansion of e.g.f. Product_{k>0} (1+tan(x)^k).
+10
6
1, 1, 2, 14, 64, 616, 5072, 58064, 669184, 9417856, 137019392, 2294104064, 40350383104, 778782954496, 16050760435712, 352024447115264, 8269739647565824, 204097141026881536, 5360540853755052032, 147190808628196081664, 4270498402940171321344, 129024432217526266494976
FORMULA
E.g.f.: exp( Sum_{k>0} (-tan(x))^k/(k*(tan(x)^k-1)) ).
MATHEMATICA
nmax = 25; CoefficientList[Series[Product[1 + Tan[x]^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 03 2020 *)
PROG
(PARI) N=40; x='x+O('x^N); Vec(serlaplace(eta(tan(x)^2)/eta(tan(x))))
(PARI) N=40; x='x+O('x^N); Vec(serlaplace(prod(k=1, N, 1+tan(x)^k)))
(PARI) N=40; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, (-tan(x))^k/(k*(tan(x)^k-1))))))
Expansion of e.g.f. Product_{k>0} 1/(1 - tan(x)^k / k).
+10
4
1, 1, 3, 13, 80, 560, 4972, 48060, 552632, 6813560, 95846728, 1435488184, 23855755040, 419889384096, 8048166402304, 162616435301824, 3531256457687168, 80497793591765120, 1953028123616286592, 49561115477458450560, 1328614915154244276224, 37134707962379971432448
FORMULA
E.g.f.: exp( Sum_{i>0} Sum_{j>0} tan(x)^(i*j)/(i*j^i) ).
MATHEMATICA
max = 21; Range[0, max]! * CoefficientList[Series[Product[1/(1 - Tan[x]^k/k), {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Oct 03 2020 *)
PROG
(PARI) N=40; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, 1-tan(x)^k/k)))
(PARI) N=40; x='x+O('x^N); Vec(serlaplace(exp(sum(i=1, N, sum(j=1, N\i, tan(x)^(i*j)/(i*j^i))))))
Expansion of e.g.f. Product_{k>0} 1/(1 - tan(x)^k / k!).
+10
4
1, 1, 3, 12, 71, 462, 3890, 35133, 381583, 4411870, 58623990, 826335675, 12990713482, 216027857567, 3925135187017, 75217607162053, 1552186877466271, 33678081631793270, 778592124168964502, 18867293553102673343, 483291402186818709310, 12937553749692179771301, 363847628395565829224327
FORMULA
E.g.f.: exp( Sum_{i>0} Sum_{j>0} tan(x)^(i*j)/(i*(j!)^i) ).
MATHEMATICA
max = 22; Range[0, max]! * CoefficientList[Series[Product[1/(1 - Tan[x]^k/k!), {k, 1, max}], {x, 0, max}], x] (* Amiram Eldar, Oct 04 2020 *)
PROG
(PARI) N=40; x='x+O('x^N); Vec(serlaplace(1/prod(k=1, N, 1-tan(x)^k/k!)))
(PARI) N=40; x='x+O('x^N); Vec(serlaplace(exp(sum(i=1, N, sum(j=1, N\i, tan(x)^(i*j)/(i*j!^i))))))
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