A189780 Expansion of e.g.f. 1/(1 - arcsin(x)).
1, 1, 2, 7, 32, 189, 1328, 11019, 104064, 1111641, 13166592, 172006671, 2448559104, 37814647701, 628513744896, 11201565483219, 212867324706816, 4299987047933745, 91950128086450176, 2076040931023605015, 49332990241672003584, 1231115505653454828525, 32183083119025449861120
Offset: 0
Keywords
Programs
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Mathematica
CoefficientList[Series[1/(1-ArcSin[t]), {t, 0, 100}], t] Table[ n!, {n, 0, 100}] (* Emanuele Munarini, Nov 23 2015 *) -
Maxima
a(n):=(n-1)!*sum(m*(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!;
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PARI
my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-asin(x)))) \\ Seiichi Manyama, Jun 26 2025
Formula
a(n)= (n-1)!*sum(m=1..n-1, m*(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!, n>0, a(0)=1.
a(n) ~ cos(1) * n! / (sin(1))^(n+1). - Vaclav Kotesovec, Nov 06 2014
a(n) = Sum_{k=0..n} k! * A385343(n,k). - Seiichi Manyama, Jun 26 2025
Extensions
More terms from Seiichi Manyama, Jun 26 2025