A270665
E.g.f.: Product_{k>=1} 1/(1 - tan(x^k)).
Original entry on oeis.org
1, 1, 4, 20, 136, 1016, 10112, 102080, 1259648, 16501888, 243214592, 3792156928, 66314635264, 1201731751936, 23824296632320, 496708324364288, 11065302289285120, 257749924759273472, 6397599337673523200, 165009476729535463424, 4496775223731602063360
Offset: 0
-
nmax = 25; Range[0, nmax]! * CoefficientList[Series[Product[1/(1-Tan[x^k]), {k, 1, nmax}], {x, 0, nmax}], x]
A330514
Expansion of e.g.f. Product_{k>=1} 1 / (1 - sin(x^k)).
Original entry on oeis.org
1, 1, 4, 17, 112, 761, 6992, 65267, 749264, 8952097, 123035312, 1765177435, 28465913320, 475981018033, 8737060100680, 167186734385795, 3446660462332576, 73894280818392641, 1691674707666258848, 40160865451008020651, 1009283508170762388536
Offset: 0
Cf.
A000111,
A203716,
A229263,
A270294,
A270662,
A270663,
A270664,
A270665,
A270666,
A330515,
A330516,
A330517.
-
nmax = 20; CoefficientList[Series[Product[1/(1 - Sin[x^k]), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
A330515
Expansion of e.g.f. Product_{k>=1} 1 / (1 - sinh(x^k)).
Original entry on oeis.org
1, 1, 4, 19, 128, 921, 8912, 87109, 1045200, 13195681, 188639312, 2837096637, 47976425576, 837845855185, 16039578298200, 321739841159317, 6911395312352672, 154749452408120385, 3696709758990757856, 91546190261460505453, 2397650607409036823352
Offset: 0
Cf.
A006154,
A203716,
A229263,
A270294,
A270662,
A270663,
A270664,
A270665,
A270666,
A330514,
A330516,
A330517.
-
nmax = 20; CoefficientList[Series[Product[1/(1 - Sinh[x^k]), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
A330516
Expansion of e.g.f. Product_{k>=1} sec(x^k) (even powers only).
Original entry on oeis.org
1, 1, 17, 601, 44225, 4589041, 781157585, 162882093193, 48519650017025, 17223202538504161, 7898449818361655825, 4193448664548573675961, 2779065418077990268214465, 2061320859693223620523895761, 1836094285018667246330440863185
Offset: 0
Cf.
A000364,
A203716,
A229263,
A270294,
A270662,
A270663,
A270664,
A270665,
A270666,
A330514,
A330515,
A330517.
-
nmax = 14; Table[(CoefficientList[Series[Product[Sec[x^k], {k, 1, nmax}], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
A330517
Expansion of e.g.f. Product_{k>=1} sech(x^k) (even powers only).
Original entry on oeis.org
1, -1, -7, -241, -4495, -652801, -15004375, -7047990769, 1597056262625, -360304327144321, 286464442762907225, 560117092794518159, 78257061390674957994065, 5684812583023438995911039, 45666128878264725133259682185
Offset: 0
Cf.
A000364,
A203716,
A229263,
A270294,
A270662,
A270663,
A270664,
A270665,
A270666,
A330514,
A330515,
A330516.
-
nmax = 14; Table[(CoefficientList[Series[Product[Sech[x^k], {k, 1, nmax}], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
A203709
E.g.f.: 2*Product_{n>=1} ((exp(x^n) + 1)/2).
Original entry on oeis.org
2, 1, 3, 10, 55, 311, 2446, 19447, 196337, 2014777, 24828706, 311108051, 4507990477, 66719239237, 1112079627842, 18945126606421, 356368711926481, 6867187345103057, 143985206958508162, 3092256807348721807, 71426909592196938101, 1691486262041519369581
Offset: 0
E.g.f.: A(x) = 2 + x + 3*x^2/2! + 10*x^3/3! + 55*x^4/4! + 311*x^5/5! +...
where
A(x) = 2*(exp(x)+1)/2 * (exp(x^2)+1)/2 * (exp(x^3)+1)/2 * (exp(x^4)+1)/2 *...
The log of the e.g.f. begins:
log(A(x)/2) = (x/2)/(1-x^2) + 5*(x/2)^2/2! + 238*(x/2)^4/4! + 28816*(x/2)^6/6! + 6397168*(x/2)^8/8! + 2322439936*(x/2)^10/10! +...
-
nmax = 25; Range[0, nmax]! * CoefficientList[Series[2*Product[1/(1 - Tanh[x^k/2]), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 21 2016 *)
-
{a(n)=n!*polcoeff(2*prod(k=1,n,(exp(x^k+x*O(x^n))+1)/2),n)}
A203715
E.g.f.: Sum_{n>=1} log((1 + exp(2*x^n))/2).
Original entry on oeis.org
1, 3, 6, 34, 120, 1096, 5040, 56848, 362880, 5451136, 39916800, 688876288, 6227020800, 130789805056, 1307674368000, 29497569445888, 355687428096000, 9746045395173376, 121645100408832000, 3451902721622867968, 51090942171709440000, 1686006043164464644096
Offset: 1
E.g.f.: A(x) = x + 3*x^2/2! + x^3 + 34*x^4/4! + x^5 + 1096*x^6/6! + x^7 + 56848*x^8/8! + x^9 + 5451136*x^10/10! + x^11 +...
where A(x) = log((1+exp(2*x))/2) + log((1+exp(2*x^2))/2) + log((1+exp(2*x^3))/2) + log((1+exp(2*x^4))/2) +...
The exponentiation of the e.g.f. begins:
exp(A(x)) = 1 + x + 4*x^2/2! + 16*x^3/3! + 104*x^4/4! + 696*x^5/5! + 6272*x^6/6! + 57856*x^7/7! + 652416*x^8/8! +...+ A203716(n)*x^n/n! +...
-
nmax = 25; Rest[Range[0, nmax]! * CoefficientList[Series[Sum[Log[1/(1 - Tanh[x^k])], {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Mar 21 2016 *)
-
{a(n)=n!*polcoeff(sum(m=1,n,log((1+exp(2*x^m+x*O(x^n)))/2)),n)}
A330538
Expansion of e.g.f. Product_{k>=1} 1 / (1 - arctanh(x^k)).
Original entry on oeis.org
1, 1, 4, 20, 136, 1024, 10208, 103872, 1287424, 17001984, 252383232, 3971122176, 69973813248, 1281307791360, 25646098022400, 540521184067584, 12170993370955776, 286967313311858688, 7206454521728335872, 188326525698968518656
Offset: 0
-
nmax = 19; CoefficientList[Series[Product[1/(1 - ArcTanh[x^k]), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 19; CoefficientList[Series[Exp[Sum[Sum[ArcTanh[x^(k/d)]^d/d, {d, Divisors[k]}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
A330518
Expansion of e.g.f. Product_{k>=1} (sec(x^k) + tan(x^k)).
Original entry on oeis.org
1, 1, 3, 14, 77, 536, 4471, 41474, 437737, 5206120, 67098091, 944705662, 14495605277, 237203399044, 4162492013135, 78089687760842, 1545654292223825, 32385137447167280, 716473190874986323, 16611710217097325366, 404119023609893926405
Offset: 0
Cf.
A000111,
A203716,
A229263,
A270294,
A270662,
A270663,
A270664,
A270665,
A270666,
A330514,
A330515,
A330516,
A330517.
-
nmax = 20; CoefficientList[Series[Product[(Sec[x^k] + Tan[x^k]), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
Showing 1-9 of 9 results.