A012150 Expansion of e.g.f. exp(tan(arcsin(x))).

1, 1, 1, 4, 13, 76, 421, 3256, 25369, 245008, 2449801, 28441216, 346065061, 4700478784, 67243537453, 1047088053376, 17192488230961, 302112622479616, 5593309059948049, 109527844826856448, 2255588021494237501

Offset

0,4

Links

Table of n, a(n) for n=0..20.

Vladimir Kruchinin and D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.

Formula

From Vladimir Kruchinin, Feb 17 2011: (Start)

a(n) = n!*Sum_{k=1..n} A111959(n-1,k-1)*2^(k-n)/k!.

a(n) = n!*Sum_{k=1..n} (1+(-1)^(n-k))*C((n-2)/2,(n-k)/2)/(2*k!), n>0.

E.g.f.: exp(x/sqrt(1-x^2)). (End)

E.g.f.: S(x) = exp(x/sqrt(1-x^2)) = 1 + 2*(x/sqrt(1-x^2))/(G(0) - x/sqrt(1-x^2)), G(k) = 8*k + 2 + (x^2)/((1-x^2)*(8*k+6) + x^2/G(k+1)); (continued fraction). - Sergei N. Gladkovskii, Dec 16 2011

a(n) = (3*n^2 - 12*n + 13)*a(n-2) - 3*(n-4)*(n-3)^2*(n-2)*a(n-4) + (n-6)*(n-5)*(n-4)^2*(n-3)*(n-2)*a(n-6). - Vaclav Kotesovec, Nov 08 2013

a(n) ~ n^(n-1/3) * exp(3/2*n^(1/3)-n) / sqrt(3) * (1 - 19/(36*n^(1/3)) + 553/(2592*n^(2/3))). - Vaclav Kotesovec, Nov 08 2013

Example

exp(tan(arcsin(x))) = 1+x+1/2!*x^2+4/3!*x^3+13/4!*x^4+76/5!*x^5...

Maple

A012150 := proc(n) if n = 0 then 1; else add( (1+(-1)^(n-k)) *binomial((n-2)/2, (n-k)/2)/(2*k!), k=1..n) ; %*n! ; end if; end proc: # R. J. Mathar, Mar 20 2011

Mathematica

Range[0, 20]! CoefficientList[Series[Exp[Tan[ArcSin[x]]], {x, 0, 20}], x] (* Or *)
f[n_] := n! Sum[(1 + (-1)^(n - k)) Binomial[(n - 2)/2, (n - k)/2]/2/k!, {k, n}]; f[0] = 1; Array[f, 21, 0] (* Robert G. Wilson v, Feb 19 2011 *)

Prog

(PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(tan(asin(x))))) \\ Michel Marcus, Oct 30 2022

Crossrefs

Cf. A111959.

Sequence in context: A171756 A235385 A144055 * A362572 A012261 A012075

Adjacent sequences: A012147 A012148 A012149 * A012151 A012152 A012153

Keyword

nonn

Author

Patrick Demichel (patrick.demichel(AT)hp.com)

Extensions

Name edited by Michel Marcus, Oct 30 2022

Status

approved