A177700
The n-th derivative of log(1+x)*tanh(x) evaluated at x = 0.
Original entry on oeis.org
0, 0, 2, -3, 0, -10, 160, -756, 2688, -27504, 341248, -3113440, 29004800, -365574144, 5120567296, -69912541440, 1009388355584, -16301637449728, 281310403362816, -5030932957138944, 94747161802047488, -1897026741117419520
Offset: 0
The second derivative is -(tanh(x)/(x+1)^2) + 2*((1 - tanh(x)^2)/(x+1)) - 2*log(x+1)tanh(x)(1 - tanh(x)^2). At x = 0 this sets a(2) = 0 + 2 - 0 = 2.
- L. Comtet and M. Fiolet, Sur les dérivées successives d'une fonction implicite. C. R. Acad. Sci. Paris Ser. A 278 (1974), 249-251. MR0348055
a(0) inserted and keyword:sign added by
R. J. Mathar, May 14 2010
A302610
Expansion of e.g.f. -log(1 - x)*arcsinh(x).
Original entry on oeis.org
0, 0, 2, 3, 4, 20, 158, 819, 3624, 33984, 427482, 3819915, 29665260, 404822340, 6948032310, 88407058635, 991515848400, 17715286764000, 383952670412850, 6349179054589875, 93532380775766100, 2063197602667372500, 53913667654307868750, 1098018631195048591875
Offset: 0
-log(1 - x)*arcsinh(x) = 2*x^2/2! + 3*x^3/3! + 4*x^4/4! + 20*x^5/5! + 158*x^6/6! + 819*x^7/7! + 3624*x^8/8! + ...
Cf.
A009410,
A009416,
A009429,
A009435,
A012572,
A104150,
A177699,
A177700,
A296435,
A296727,
A302611.
-
a:=series(-log(1-x)*arcsinh(x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 26 2019
-
nmax = 23; CoefficientList[Series[-Log[1 - x] ArcSinh[x], {x, 0, nmax}], x] Range[0, nmax]!
A302611
Expansion of e.g.f. -log(1 - x)*arctanh(x).
Original entry on oeis.org
0, 0, 2, 3, 16, 50, 368, 1764, 16896, 109584, 1297152, 10628640, 149944320, 1486442880, 24349317120, 283465647360, 5287713177600, 70734282393600, 1480103564083200, 22376988058521600, 519000166327910400, 8752948036761600000, 222845873874075648000, 4148476779335454720000
Offset: 0
-log(1 - x)*arctanh(x) = 2*x^2/2! + 3*x^3/3! + 16*x^4/4! + 50*x^5/5! + 368*x^6/6! + 1764*x^7/7! + 16896*x^8/8! + ...
Cf.
A005359,
A009410,
A009416,
A009429,
A009435,
A012697,
A081358,
A104150,
A177699,
A177700,
A202139,
A302610.
-
a:=series(-log(1-x)*arctanh(x),x=0,24): seq(n!*coeff(a,x,n),n=0..23); # Paolo P. Lava, Mar 26 2019
-
nmax = 23; CoefficientList[Series[-Log[1 - x] ArcTanh[x], {x, 0, nmax}], x] Range[0, nmax]!
-
x='x+O('x^99); concat([0, 0], Vec(serlaplace(log(1-x)*log((1-x)/(1+x))/2))) \\ Altug Alkan, Apr 10 2018
A381016
Expansion of e.g.f. -log(1-x) * sin(x).
Original entry on oeis.org
0, 0, 2, 3, 4, 20, 110, 651, 4520, 36000, 322618, 3213595, 35226860, 421419492, 5463436134, 76301056755, 1142009233872, 18236159031584, 309463272791538, 5561354285804115, 105510576441518164, 2107380222724155540, 44200537412519181278, 971311172969442165883
Offset: 0
-
my(N=30, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(-log(1-x)*sin(x))))
-
a(n) = -sum(k=1, n\2, (-1)^k*(n-2*k)!*binomial(n, 2*k-1));
Showing 1-4 of 4 results.