A189815 Expansion of e.g.f. log(1/(1-arcsin(x))).

1, 1, 3, 10, 53, 304, 2303, 18768, 185033, 1954176, 23756667, 308077056, 4457821821, 68513332224, 1150764459063, 20443736745984, 391167511473681, 7884821722497024, 169370797497060339, 3818539009013907456, 91013260219635394629, 2269047587255097753600, 59435772666287730632559

Offset

1,3

Links

G. C. Greubel, Table of n, a(n) for n = 1..435

Formula

a(n) = (n-1)!*Sum_{m=1..(n-1)} ((1+(-1)^(n-m))/2)*Sum_{k=1..(n-m)} (Sum_{j=1..k} binomial(k,j)*2^(1-j)*Sum_{i=0..floor(j/2)} (-1)^((n-m)/2-i-j)*binomial(j,i)*(j-2*i)^(n-m+j)/(n-m+j)!)*binomial(k+n-1,n-1)) + (n-1)!.

a(n) ~ (n-1)! / (sin(1))^n. - Vaclav Kotesovec, Nov 06 2014

Maple

a:=series(log(1/(1-arcsin(x))), x=0, 24): seq(n!*coeff(a, x, n), n=1..23); # Paolo P. Lava, Mar 27 2019

Mathematica

Rest[With[{nmax = 50}, CoefficientList[Series[Log[1/(1 - ArcSin[x])], {x, 0, nmax}], x]*Range[0, nmax]!]] (* G. C. Greubel, Jan 16 2018 *)

Prog

(Maxima) a(n):=(n-1)!*sum((1+(-1)^(n-m))/2*sum((sum(binomial(k, j)*2^(1-j)*sum((-1)^((n-m)/2-i-j)*binomial(j, i)*(j-2*i)^(n-m+j)/(n-m+j)!, i, 0, floor(j/2)), j, 1, k))*binomial(k+n-1, n-1), k, 1, n-m), m, 1, n-1)+(n-1)!;
(PARI) x='x+O('x^30); Vec(serlaplace(log(1/(1-asin(x))))) \\ G. C. Greubel, Jan 16 2018

Crossrefs

Sequence in context: A309910 A042171 A133148 * A143599 A264409 A199202

Adjacent sequences: A189812 A189813 A189814 * A189816 A189817 A189818

Keyword

nonn

Author

Vladimir Kruchinin, May 02 2011

Extensions

Terms a(18) onward added by G. C. Greubel, Jan 16 2018

Status

approved