A191301 Expansion of exp(x*arcsin(x)) = sum_{n>=0} a(n)*x^(2n)/product(k, 0<k<2n).

1, 1, 4, 49, 1303, 63513, 5044601, 598488981, 99463522845, 22073876512113, 6308016788410641, 2256148067062888845, 987271755178677563541, 518851042593331302909225, 322466959923499314299129625, 233940983258826325064978994501, 195913817323275641425583595611805

Offset

0,3

Comments

Expansion of exp(sqrt(x)*arcsin(sqrt(x))) = sum_{n>=0} a(n)*x^(n)/product(k, 0<k<2n). - Daniel Forgues, Mar 05 2013

Links

Alois P. Heinz, Table of n, a(n) for n = 0..100

Formula

a(n) = (2*n-1)!*sum(m=1..n, sum(k=0..n, (-1)^k*C((2*n-m-2)/2,n-m-k)* sum(i=0..2*k, (2^i*stirling1(i+m,m)*C(2*k+m-1,i+m-1))/(i+m)!))), n>0, a(0)=1.

a(n) ~ exp(Pi/2) * 2^(2*n-1) * n^(2*n-2) / exp(2*n). - Vaclav Kotesovec, Apr 05 2016

Example

exp(x*arcsin(x)) =
  1 + 1/1!*x^2 + 4/3!*x^4 + 49/5!*x^6 + 1303/7!*x^8 + 63513/9!*x^10 + ...
exp(sqrt(x)*arcsin(sqrt(x))) =
  1 + 1/1!*x^1 + 4/3!*x^2 + 49/5!*x^3 + 1303/7!*x^4 + 63513/9!*x^5 + ...

Mathematica

a[n_] := (2*n-1)!* Sum[ Sum[ (Sum[ (2^i*StirlingS1[i+m, m]* Binomial[2*k+m-1, i+m-1])/(i+m)!, {i, 0, 2*k}])*(-1)^k*Binomial[(2*n-m-2)/2, n-m-k], {k, 0, n}], {m, 1, n}]; a[0] = 1; Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Mar 04 2013, translated from Maxima *)

Prog

(Maxima)
a(n):=(2*n-1)!*sum(sum((sum((2^i*stirling1(i+m, m)*binomial(2*k+m-1, i+m-1))/ (i+m)!, i, 0, 2*k))*(-1)^k*binomial((2*n-m-2)/2, n-m-k), k, 0, n), m, 1, n);

Crossrefs

Cf. A042972.

Sequence in context: A329328 A188682 A354865 * A336805 A029991 A129419

Adjacent sequences: A191298 A191299 A191300 * A191302 A191303 A191304

Keyword

nonn

Author

Vladimir Kruchinin, May 30 2011

Status

approved