A177381 G.f.: Sum_{n>=0} Product_{k=1..n} tan(k*arctan(x)).
1, 1, 2, 6, 26, 142, 930, 7110, 62138, 610958, 6674370, 80201222, 1051277530, 14927729678, 228262465634, 3739557703366, 65345926588026, 1213197344607502, 23848186328994178, 494822251631023622, 10807111342480752538
Offset: 0
Keywords
Examples
G.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 26*x^4 + 142*x^5 + 930*x^6 +... A(x) = 1 + x + x*tan(2*arctan(x)) + x*tan(2*arctan(x))*tan(3*arctan(x)) +... where the series tan(k*arctan(x)) for k=2..6 begin: tan(2*arctan(x)) = 2*x + 2*x^3 + 2*x^5 + 2*x^7 + 2*x^9 + ... tan(3*arctan(x)) = 3*x + 8*x^3 + 24*x^5 + 72*x^7 + 216*x^9 + ... tan(4*arctan(x)) = 4*x + 20*x^3 + 116*x^5 + 676*x^7 + 3940*x^9 + ... tan(5*arctan(x)) = 5*x + 40*x^3 + 376*x^5 + 3560*x^7 + 33720*x^9 + ... tan(6*arctan(x)) = 6*x + 70*x^3 + 966*x^5 + 13446*x^7 + 187270*x^9 + ... ... tan(k*arctan(x)) = -i*((1+i*x)^k - (1-i*x)^k) / ((1+i*x)^k + (1-i*x)^k).
Links
- Vaclav Kotesovec, Table of n, a(n) for n = 0..300
Programs
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PARI
{a(n)=local(X=x+x*O(x^n),Gf);Gf=sum(m=0,n,prod(k=1,m,tan(k*atan(X))));polcoeff(Gf,n)} -
PARI
{a(n)=polcoeff(sum(m=0, n, (-I)^m*prod(k=1, m, ((1+I*x)^k-(1-I*x)^k)/((1+I*x)^k+(1-I*x)^k +x*O(x^n)))), n)} for(n=0,25,print1(a(n),","))
Formula
G.f.: A(x) = G(arctan(x)) where G(x) = e.g.f. of A177382.
G.f.: Sum_{n>=0} (-I)^n*Product_{k=1..n} ((1+i*x)^k - (1-i*x)^k)/((1+i*x)^k + (1-i*x)^k), where i = sqrt(-1).
a(n) ~ n! / (sqrt(2) * G^(n+1)), where G = A006752 = 0.915965594177219... is Catalan's constant. - Vaclav Kotesovec, Nov 06 2014