A199569 Expansion 1/(1-x^2*cosech(x)) = Sum_{n>=0} a(n)*x^n/n!^2.

1, 1, 4, 30, 384, 7480, 207360, 7780080, 380190720, 23481311616, 1789201612800, 164904696633600, 18084647927808000, 2327418985883397120, 347368297708734382080, 59514548453599599360000, 11601363342443780505600000, 2552998389393196650531225600

Offset

0,3

Links

Table of n, a(n) for n=0..17.

Formula

a(n) = n!^2*sum(m=1..n, m!*sum(i=0..n-m, (2^i*m^(n-m-i)* sum(k=0..i, (Stirling2(i,k)*k!*Stirling1(m+k,m))/(m+k)!))/(i!*(n-m-i)!))), n>0, a(0)=1.

a(n) ~ n!^2 * sinh(r)^2 / ((2*sinh(r) - r*cosh(r)) * r^(n+2)), where r = 1.3132837183534835944436309473975491200703574291896572634517... is the root of the equation 2*r^2 = exp(r) - exp(-r). - Vaclav Kotesovec, Jan 23 2025

Example

1/(1-x^2*csch(x)) = 1 + x + x^2 + (5*x^3)/6 + (2*x^4)/3 + (187*x^5)/360 + (2*x^6)/5 + (4631*x^7)/1512 + (221*x^8)/945 + (11983*x^9)/67200 + (214*x^10)/1575 + ...

Mathematica

nmax = 20; CoefficientList[Series[1/(1-x^2*Csch[x]), {x, 0, nmax}], x] * Range[0, nmax]!^2 (* Vaclav Kotesovec, Jan 23 2025 *)

Prog

(Maxima) a(n) := if n=0 then 1 else n!^2 * sum(m!*sum((2^i*m^(n-m-i)* sum((stirling2(i, k)*k!*stirling1(m+k, m))/(m+k)!, k, 0, i))/(i!*(n-m-i)!), i, 0, n-m), m, 1, n);

Crossrefs

Sequence in context: A336712 A185523 A187736 * A363911 A326205 A351795

Adjacent sequences: A199566 A199567 A199568 * A199570 A199571 A199572

Keyword

nonn

Author

Vladimir Kruchinin, Nov 08 2011

Status

approved