A331615
E.g.f.: exp(1 / (1 - arcsin(x)) - 1).
Original entry on oeis.org
1, 1, 3, 14, 85, 640, 5703, 58760, 685353, 8925632, 128231627, 2014061568, 34312150525, 630043097216, 12400033125647, 260357810321664, 5807790344591953, 137144754146230272, 3417248676737769619, 89590823377278496768, 2465026658283881339301
Offset: 0
-
nmax = 20; CoefficientList[Series[Exp[1/(1 - ArcSin[x]) - 1], {x, 0, nmax}], x] Range[0, nmax]!
A189780[0] = 1; A189780[n_] := A189780[n] = Sum[Binomial[n, k] If[OddQ[k], ((k - 2)!!)^2, 0] A189780[n - k], {k, 1, n}]; a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] A189780[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
-
seq(n)={Vec(serlaplace(exp(1/(1 - asin(x + O(x*x^n))) - 1)))} \\ Andrew Howroyd, Jan 22 2020
A013170
Expansion of e.g.f.: exp(arctanh(x)+arcsin(x)).
Original entry on oeis.org
1, 2, 4, 11, 40, 185, 1030, 6785, 51160, 438425, 4176250, 44098925, 507867100, 6375156125, 86130937750, 1253044612625, 19429666282000, 321457635604625, 5626346310003250, 104323833636303125, 2034312191276462500
Offset: 0
Patrick Demichel (patrick.demichel(AT)hp.com)
exp(arctanh(x)+arcsin(x)) = 1+2*x+4/2!*x^2+11/3!*x^3+40/4!*x^4+185/5!*x^5+...
-
Table[n!*SeriesCoefficient[E^(ArcTanh[x]+ArcSin[x]),{x,0,n}],{n,0,20}] (* Vaclav Kotesovec, Oct 07 2012 *)
-
x='x+O('x^66); Vec(serlaplace(exp(atanh(x)+asin(x)))) \\ Joerg Arndt, May 04 2013
A168405
E.g.f.: Sum_{n>=0} arcsin(2^n*x)^n/n!.
Original entry on oeis.org
1, 2, 16, 520, 66560, 33882400, 69055283200, 564153087455360, 18462510039810703360, 2418626471936038215754240, 1267795676362601991645220044800, 2658560574070850656450883768752998400
Offset: 0
E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 520*x^3/3! + 66560*x^4/4! + ...
A(x) = 1 + arcsin(2*x) + arcsin(4*x)^2/2! + arcsin(8*x)^3/3! + arcsin(16*x)^4/4! + ... + arcsin(2^n*x)^n/n! + ...
a(n) = coefficient of x^n/n! in G(x)^(2^n) where G(x) = exp(arcsin(x)):
G(x) = 1 + x + x^2/2! + 2*x^3/3! + 5*x^4/4! + 20*x^5/5! + 85*x^6/6! + ... + A006228(n)*x^n/n! + ...
-
{a(n)=n!*polcoeff(sum(k=0,n,asin(2^k*x +x*O(x^n))^k/k!),n)}
-
{a(n)=n!*polcoeff(exp(2^n*asin(x +x*O(x^n))),n)}
A296788
Expansion of e.g.f. exp(x*arcsinh(x)) (even powers only).
Original entry on oeis.org
1, 2, 8, 54, 104, 18810, -1648428, 247726374, -49445941200, 12841169289714, -4206667789245780, 1697448414191239710, -827415782970517712376, 479396168140498731959850, -325673237888367403728512700, 256401822876859593450127851030, -231597610351491427264049084814240
Offset: 0
exp(x*arcsinh(x)) = 1 + 2*x^2/2! + 8*x^4/4! + 54*x^6/6! + 104*x^8/8! + ...
-
nmax = 16; Table[(CoefficientList[Series[Exp[x ArcSinh[x]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
nmax = 16; Table[(CoefficientList[Series[(x + Sqrt[1 + x^2])^x, {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]
A293191
a(n) = n! * [x^n] exp(n*arcsin(x)).
Original entry on oeis.org
1, 1, 4, 30, 320, 4420, 74880, 1502200, 34816000, 915267600, 26907545600, 874679634400, 31150846771200, 1206169954600000, 50449154129920000, 2266730426366640000, 108883850653859840000, 5568313898902866592000, 302047359993635143680000, 17321727685903810436800000
Offset: 0
-
Table[n! SeriesCoefficient[Exp[n ArcSin[x]], {x, 0, n}], {n, 0, 19}]
A013430
Expansion of e.g.f. exp(arcsin(x)-arctanh(x)).
Original entry on oeis.org
1, 0, 0, -1, 0, -15, 10, -495, 840, -29575, 87750, -2805075, 12340900, -387547875, 2304422250, -73485787375, 556725078000, -18311017515375, 169707184396750, -5801182757296875, 63859173696337500
Offset: 0
Patrick Demichel (patrick.demichel(AT)hp.com)
1-1/3!*x^3-15/5!*x^5+10/6!*x^6-495/7!*x^7...
-
With[{nn=20},CoefficientList[Series[Exp[ArcSin[x]-ArcTanh[x]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 23 2025 *)
-
my(x='x+O('x^30)); Vec(serlaplace(exp(asin(x)-atanh(x)))) \\ Michel Marcus, May 10 2020