This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A189780 #29 Jun 26 2025 08:01:05
%S A189780 1,1,2,7,32,189,1328,11019,104064,1111641,13166592,172006671,
%T A189780 2448559104,37814647701,628513744896,11201565483219,212867324706816,
%U A189780 4299987047933745,91950128086450176,2076040931023605015,49332990241672003584,1231115505653454828525,32183083119025449861120
%N A189780 Expansion of e.g.f. 1/(1 - arcsin(x)).
%F A189780 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.
%F A189780 a(n) ~ cos(1) * n! / (sin(1))^(n+1). - _Vaclav Kotesovec_, Nov 06 2014
%F A189780 a(n) = Sum_{k=0..n} k! * A385343(n,k). - _Seiichi Manyama_, Jun 26 2025
%t A189780 CoefficientList[Series[1/(1-ArcSin[t]), {t, 0, 100}], t] Table[
%t A189780 n!, {n, 0, 100}] (* _Emanuele Munarini_, Nov 23 2015 *)
%o A189780 (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!;
%o A189780 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-asin(x)))) \\ _Seiichi Manyama_, Jun 26 2025
%Y A189780 Cf. A385346, A385347.
%Y A189780 Cf. A385343.
%K A189780 nonn
%O A189780 0,3
%A A189780 _Vladimir Kruchinin_, May 02 2011
%E A189780 More terms from _Seiichi Manyama_, Jun 26 2025