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A214687 E.g.f.: Sum_{n>=0} exp(2*n*x) * Product_{k=1..n} (exp((2*k-1)*x) - 1). 2
1, 1, 11, 217, 7691, 430921, 35117531, 3927676537, 577640740331, 108115035641641, 25097054302205051, 7076531411753120857, 2382432541064412524171, 943997056642739165681161, 434864796716131476530668571, 230460477665217932140097413177 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Compare the e.g.f. to the identity:

exp(-x) = Sum_{n>=0} exp(2*n*x) * Product_{k=1..n} (1 - exp((2*k-1)*x)).

LINKS

Vaclav Kotesovec, Table of n, a(n) for n = 0..175

Hsien-Kuei Hwang, Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

FORMULA

E.g.f. A(x) satisfies: A(x) = exp(-x)*(2*G(x) - 1),

where G(x) = Sum_{n>=0} Product_{k=1..n} (exp((2*k-1)*x) - 1) = e.g.f. of A215066.

a(n) ~ 2*sqrt(6) * 24^n * (n!)^2 / (sqrt(n) * Pi^(2*n+3/2)). - Vaclav Kotesovec, May 05 2014

EXAMPLE

E.g.f.: A(x) = 1 + x + 11*x^2/2! + 217*x^3/3! + 7691*x^4/4! + 430921*x^5/5! +...

such that, by definition,

A(x) = 1 + exp(2*x)*(exp(x)-1) + exp(4*x)*(exp(x)-1)*(exp(3*x)-1)

+ exp(6*x)*(exp(x)-1)*(exp(3*x)-1)*(exp(5*x)-1)

+ exp(8*x)*(exp(x)-1)*(exp(3*x)-1)*(exp(5*x)-1)*(exp(7*x)-1) +...

Compare this series to the identity:

exp(-x) = 1 - exp(2*x)*(exp(x)-1) + exp(4*x)*(exp(x)-1)*(exp(3*x)-1)

- exp(6*x)*(exp(x)-1)*(exp(3*x)-1)*(exp(5*x)-1)

+ exp(8*x)*(exp(x)-1)*(exp(3*x)-1)*(exp(5*x)-1)*(exp(7*x)-1)  +-...

The related e.g.f. of A215066 equals the series:

G(x) = 1 + (exp(x)-1) + (exp(x)-1)*(exp(3*x)-1)

+ (exp(x)-1)*(exp(3*x)-1)*(exp(5*x)-1)

+ (exp(x)-1)*(exp(3*x)-1)*(exp(5*x)-1)*(exp(7*x)-1) +...

or, more explicitly,

G(x) = 1 + x + 7*x^2/2! + 127*x^3/3! + 4315*x^4/4! + 235831*x^5/5! +...

such that G(x) satisfies:

G(x) = (1 + exp(x)*A(x))/2.

PROG

(PARI) {a(n)=n!*polcoeff(sum(m=0, n+1, exp(2*m*x+x*O(x^n))*prod(k=1, m, exp((2*k-1)*x+x*O(x^n))-1)), n)}

for(n=0, 26, print1(a(n), ", "))

CROSSREFS

Cf. A207214, A215066.

Sequence in context: A134069 A137464 A187650 * A160074 A298889 A204236

Adjacent sequences:  A214684 A214685 A214686 * A214688 A214689 A214690

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Aug 01 2012

STATUS

approved

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Last modified February 26 17:04 EST 2021. Contains 341632 sequences. (Running on oeis4.)