A003716 -id:A003716 - cp's OEIS frontend

A003721 Expansion of e.g.f. tan(tanh(x)) (odd powers only).

1, 0, -8, 112, -128, -109824, 8141824, -353878016, -9666461696, 5151942574080, -825073851170816, 76429076694827008, 2051308253366714368, -2361338488910424047616, 171581865952588387581952

Offset: 0

R. H. Hardin, Simon Plouffe

sign

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..100

Cf. A003716, A003718.

With[{nn=30},Take[CoefficientList[Series[Tan[Tanh[x]],{x,0,nn}],x] Range[0,nn-1]!,{2,-1,2}]] (* Harvey P. Dale, Nov 05 2011 *)

a(n):=sum(((sum(j!*2^(2*i+1-j-1)*(-1)^(i+j+1)*stirling2(2*i+1,j),j,1,2*i+1))*sum(binomial(k-1,2*i)*k!*(-1)^(1+k)*2^(2*n-k-1)*stirling2(2*n-1,k),k,2*i+1,2*n-1))/(2*i+1)!,i,0,(n-1)); /* Vladimir Kruchinin, Jun 10 2011 */

a(n) = Sum_{i=0..(n-1)} ( ( Sum_{j=1..2*i+1} j!*2^(2*i+1-j-1)*(-1)^(i+j+1)*Stirling2(2*i+1,j) ) * Sum_{k=2*i+1..2*n-1} binomial(k-1,2*i)*k!*(-1)^(1+k)*2^(2*n-k-1)*Stirling2(2*n-1,k) )/(2*i+1)!. - Vladimir Kruchinin, Jun 10 2011

A296678 Expansion of e.g.f. arctanh(arcsin(x)) (odd powers only).

Original entry on oeis.org

1, 3, 53, 2303, 185033, 23756667, 4457821821, 1150764459063, 391167511473681, 169370797497060339, 91013260219635394629, 59435772666287730632559, 46362471059282707504957401, 42577231265939498962852834155, 45471686987452309473064526678925

Offset: 0

Ilya Gutkovskiy, Dec 18 2017

nonn

			arctanh(arcsin(x)) = x/1! + 3*x^3/3! + 53*x^5/5! + 2303*x^7/7! + 185033*x^9/9! + 23756667*x^11/11! + ...

Cf. A001818, A003716, A010050, A012258, A296465, A296467, A296677.

nmax = 15; Table[(CoefficientList[Series[ArcTanh[ArcSin[x]], {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]
nmax = 15; Table[(CoefficientList[Series[Log[1 - I Log[I x + Sqrt[1 - x^2]]]/2 - Log[1 + I Log[I x + Sqrt[1 - x^2]]]/2, {x, 0, 2 nmax + 1}], x] Range[0, 2 nmax + 1]!)[[n]], {n, 2, 2 nmax, 2}]

E.g.f.: arctan(arcsinh(x)) (odd powers only, absolute values).

E.g.f.: log(1 - i*log(i*x + sqrt(1 - x^2)))/2 - log(1 + i*log(i*x + sqrt(1 - x^2)))/2, where i is the imaginary unit (odd powers only).

cp's OEIS Frontend

A003721 Expansion of e.g.f. tan(tanh(x)) (odd powers only).

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