A177389 Expansion of o.g.f.: Sum_{n>=0} Product_{k=1..n} tanh(k*arctanh(x)).

1, 1, 2, 6, 22, 98, 514, 3110, 21334, 163650, 1388162, 12902086, 130391830, 1423632546, 16699055490, 209432697830, 2796597560150, 39613075175554, 593253347702530, 9366042608039814, 155466234198142998

Offset: 0

Author

Paul D. Hanna, May 15 2010

Keywords

• nonn

Examples

			O.g.f.: A(x) = 1 + x + 2*x^2 + 6*x^3 + 22*x^4 + 98*x^5 + 514*x^6 + ...
Let G(x) = Sum_{n>=0} A002105(n+1)*x^n/n!, so that
G(x) = 1 + x + 4*x^2/2! + 34*x^3/3! + 496*x^4/4! + 11056*x^5/5! + ...
then A(x) = G(arctanh(x)).
G.f.: 1 + x + x*(2x/(1+x^2)) + x*(2x/(1+x^2))*((3x+x^3)/(1+3x^2)) + x*(2x/(1+x^2))*((3x+x^3)/(1+3x^2))*((4x+4x^3)/(1+6x^2+x^4)) + ... - _Paul D. Hanna_, May 22 2010
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..280

Crossrefs

Cf. A002105.

Programs

PARI

{a(n)=local(X=x+x*O(x^n),Egf);Egf=sum(m=0,n,prod(k=1,m,tanh(k*atanh(X))));polcoeff(Egf,n)}

PARI

{a(n)=polcoeff(sum(m=0,n,prod(k=1,m,((1+x)^k-(1-x)^k)/((1+x)^k+(1-x)^k+x*O(x^n)))),n)} \\ Paul D. Hanna, May 22 2010

Formula

O.g.f.: A(x) = Sum_{n>=0} A002105(n+1)*arctanh(x)^n/n!, where A002105 is the reduced tangent numbers.

G.f.: Sum_{n>=0} Product_{k=1..n} ((1+x)^k - (1-x)^k)/((1+x)^k + (1-x)^k). - Paul D. Hanna, May 22 2010

a(n) ~ 2^(3*n+9/2) * n^(n+1) / (exp(n) * Pi^(2*n+2)). - Vaclav Kotesovec, Nov 06 2014