A203716
E.g.f.: Product_{n>=1} (exp(2*x^n) + 1)/2.
Original entry on oeis.org
1, 1, 4, 16, 104, 696, 6272, 57856, 652416, 7657600, 104244992, 1475430144, 23426373632, 387521615872, 7034561925120, 132850810138624, 2709375373672448, 57456525327335424, 1301169515685085184, 30573796812553584640, 760486440376336908288, 19600568102376899608576
Offset: 0
E.g.f.: A(x) = 1 + x + 4*x^2/2! + 16*x^3/3! + 104*x^4/4! + 696*x^5/5! +...
where the e.g.f. equals the product:
A(x) = (exp(2*x)+1)/2 * (exp(2*x^2)+1)/2 * (exp(2*x^3)+1)/2 * (exp(2*x^4)+1)/2 *...
The log of the e.g.f. begins:
log(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 +...+ A203715(n)*x^n/n! +...
Note that the coefficients of the odd powers of x in log(A(x)) equals 1.
-
nmax = 25; Range[0, nmax]! * CoefficientList[Series[Product[1/(1 - Tanh[x^k]), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 21 2016 *)
-
{a(n)=n!*polcoeff(prod(k=1, n, (exp(2*x^k+x*O(x^n))+1)/2), n)}
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]
A270666
E.g.f.: Product_{k>=1} (1 + tan(x^k)).
Original entry on oeis.org
1, 1, 2, 14, 48, 416, 3360, 29504, 274176, 3503104, 45192960, 579956992, 8982251520, 138130720768, 2456648183808, 45868468109312, 871166211686400, 17536583860060160, 393972064172900352, 8704569607311982592, 210657904645299240960, 5322004254737369399296
Offset: 0
-
nmax = 25; Range[0, nmax]! * CoefficientList[Series[Product[(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}]
A330544
Expansion of e.g.f. Product_{k>=1} (1 + arctanh(x^k)).
Original entry on oeis.org
1, 1, 2, 14, 48, 424, 3360, 30288, 276864, 3591936, 46241280, 599212800, 9205954560, 142744412160, 2554915184640, 47649718609920, 907617573273600, 18296536869273600, 413470794456760320, 9130651338347642880, 221996730181563187200
Offset: 0
-
nmax = 20; CoefficientList[Series[Product[(1 + ArcTanh[x^k]), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 20; CoefficientList[Series[Exp[Sum[Sum[(-1)^(d + 1) 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.