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URL: https://oeis.org/A177382

⇱ A177382 - OEIS

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A177382
E.g.f.: Sum_{n>=0} Product_{k=1..n} tan(k*x).
3
1, 1, 4, 38, 656, 17776, 695104, 37049648, 2581673216, 227817246976, 24829660693504, 3275474443371008, 514345822537650176, 94806411271686270976, 20269838348763427323904, 4975513260049237751994368
OFFSET
0,3
COMMENTS
Compare to an e.g.f. of A000182, the tangent numbers:
Sum_{n>=0} A000182(n)*x^n/n! = Sum_{n>=0} Product_{k=1..n} tanh(k*x).
LINKS
FORMULA
From Vaclav Kotesovec, Nov 02 2014: (Start)
a(n) ~ (n!)^2 / (sqrt(2) * G^(n+1)).
a(n) ~ Pi * sqrt(2) * n^(2*n+1) / (exp(2*n) * G^(n+1)), where G = A006752 = 0.915965594177219... is Catalan's constant.
(End)
EXAMPLE
E.g.f.: 1 + x + 4*x^2/2! + 38*x^3/3! + 656*x^4/4! + 17776*x^5/5! +...
where
A(x) = 1 + tan(x) + tan(x)*tan(2*x) + tan(x)*tan(2*x)*tan(3*x) + tan(x)*tan(2*x)*tan(3*x)*tan(4*x) + tan(x)*tan(2*x)*tan(3*x)*tan(4*x)*tan(5*x) +...
MATHEMATICA
nmax = 20; CoefficientList[Series[Sum[Product[Tan[k*x], {k, 1, n}], {n, 0, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 02 2020 *)
PROG
(PARI) {a(n)=local(X=x+x*O(x^n), Egf); Egf=sum(m=0, n, prod(k=1, m, tan(k*X))); n!*polcoeff(Egf, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 11 2010
STATUS
approved