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A215890 E.g.f.: Shw(x) / Chw(x), where Chw(x) = Sum_{n>=0} (2*n+1)^(2*n-1) * x^(2*n)/(2*n)! and Shw(x) = Sum_{n>=0} (2*n+2)^(2*n) * x^(2*n+1)/(2*n+1)!, with zero coefficients omitted. 5
1, 7, 461, 84827, 30648697, 18319873199, 16364889804485, 20422587185959363, 33936856584735107441, 72442485189323515166807, 193177550094285124372564285, 629325007854898466696826469931, 2459394825443160188158170494692841, 11355850594904678701281528241280254207, 61160283986030409241528644478489460162357, 380011322281573634548774673447708463134715539 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

It appears that terms are not divisible by 2 or 3.

LINKS

Paul D. Hanna, Table of n, a(n) for n = 1..200

FORMULA

E.g.f.: tanh(x*Chw(x)) = tanh( Sum_{n>=0} (2*n+1)^(2*n)*x^(2*n+1)/(2*n+1)! ).

EXAMPLE

E.g.f.: A(x) = x + 7*x^3/3! + 461*x^5/5! + 84827*x^7/7! + 30648697*x^9/9! +...

such that A(x) = Shw(x)/Chw(x) = tanh(x*Chw(x)) where

Shw(x) = x + 16*x^3/3! + 1296*x^5/5! + 262144*x^7/7! + 100000000*x^9/9! +...+ (2*n+2)^(2*n)*x^(2*n+1)/(2*n+1)! +...

Chw(x) = 1 + 3*x^2/2! + 125*x^4/4! + 16807*x^6/6! + 4782969*x^8/8! + 2357947691*x^10/10! +...+ (2*n+1)^(2*n-1)*x^(2*n)/(2*n)! +...

and Chw(x) + Shw(x) = LambertW(-x)/(-x).

MATHEMATICA

max = 28; Chw[x_] := (-ProductLog[-x] + ProductLog[x])/(2*x); Shw[x_] := -(ProductLog[-x] + ProductLog[x])/(2*x); se = Series[ Shw[x]/Chw[x], {x, 0, max}]; (CoefficientList[se, x] // DeleteCases[#, 0] &)*Range[1, max, 2]! (* Jean-Fran├žois Alcover, Jun 24 2013 *)

PROG

(PARI) {a(n)=local(Chw=sum(m=0, n, (2*m+1)^(2*m-1)*x^(2*m)/(2*m)!)+x*O(x^(2*n)), Shw=sum(m=0, n, (2*m+2)^(2*m)*x^(2*m+1)/(2*m+1)!)+x*O(x^(2*n+1))); (2*n+1)!*polcoeff(Shw/Chw, 2*n+1)}

for(n=0, 20, print1(a(n), ", "))

(PARI) {a(n)=(2*n+1)!*polcoeff( tanh( sum(m=0, n, (2*m+1)^(2*m)*x^(2*m+1)/(2*m+1)!) +x*O(x^(2*n+1))), 2*n+1)}

for(n=0, 20, print1(a(n), ", "))

CROSSREFS

Cf. A215880, A215881, A215882, A216143.

Sequence in context: A261479 A241363 A196540 * A160374 A127908 A015105

Adjacent sequences:  A215887 A215888 A215889 * A215891 A215892 A215893

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 25 2012

STATUS

approved

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Last modified June 27 17:46 EDT 2022. Contains 354896 sequences. (Running on oeis4.)