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

1, 1, 7, 127, 4315, 235831, 18911467, 2091412807, 305035062955, 56729101908151, 13102338649018027, 3679320979659518887, 1234515698986458346795, 487763952468349266962071, 224150079034073231822617387, 118541831524545132821950527367

Offset: 0

Author

Paul D. Hanna, Aug 01 2012

Keywords

• nonn
• easy

Examples

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 127*x^3/3! + 4315*x^4/4! + 235831*x^5/5! +...
where
A(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) + (exp(x)-1)*(exp(3*x)-1)*(exp(5*x)-1)*(exp(7*x)-1)*(exp(9*x)-1) +...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..234
• A. Folsom, K. Ono and R. C. Rhoades, Ramanujan's radial limits, 2013. - From _N. J. A. Sloane_, Feb 09 2013
• Hsien-Kuei Hwang and Emma Yu Jin, Asymptotics and statistics on Fishburn matrices and their generalizations, arXiv:1911.06690 [math.CO], 2019.

Crossrefs

Cf. A053250, A158690, A207569, A214687.

Programs

Magma

m:=20; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( (&+[(&*[Exp((2*k-1)*x) -1: k in [1..j]]): j in [1..m+1]]) )); [1] cat [Factorial(n)*b[n]: n in [1..m-1]]; // G. C. Greubel, Feb 07 2020

Maple

m:= 20; S:= series( add(mul(exp((2*k-1)*x)-1, k=1..j), j=0..m+1), x, m+1): seq(factorial(j)*coeff(S, x, j), j = 0..m); # G. C. Greubel, Feb 07 2020

Mathematica

Table[((-1)^n*2*Sum[Sum[n!/(a!*(2b)!*(n-a-2b)!)*(3/2)^a*(5/2)^(2b) * EulerE[2a+2b],{a,0,n}],{b,0,n/2}] + 2*(-1)^n*Sum[n!/((n-2b)!*(2b)!)*(3/2)^(n-2b)*(1/2)^(2b)*EulerE[2n-2b],{b,0,n/2}])/4,{n,0,20}] (* Vaclav Kotesovec, May 04 2014 after A. Folsom *)
With[{m=20}, CoefficientList[Series[Sum[Product[Exp[(2*k-1)*x] -1, {k, j}], {j, 0, m+2}], {x,0,m}], x]*Range[0, m]!] (* G. C. Greubel, Feb 07 2020 *)

PARI

{a(n)=n!*polcoeff(sum(m=0, n+1, prod(k=1, m, exp((2*k-1)*x+x*O(x^n))-1)), n)}
for(n=0, 26, print1(a(n), ", "))

Sage

m=20;
def A215066_list(prec):
    P. = PowerSeriesRing(QQ, prec)
    return P( sum(product(exp((2*k-1)*x)-1 for k in (1..j)) for j in (0..m)) ).list()
a=A215066_list(m+1); [factorial(n)*a[n] for n in (0..m)] # G. C. Greubel, Feb 07 2020

Formula

Folsom et al. give a closed form for a(n). - N. J. A. Sloane, Feb 09 2013

E.g.f.: 1 + (exp(x)-1)/(W(0)-exp(x)+1), where W(k) = (exp(x))^(2*k+1) - ((exp(x))^(2*k+3)-1)/W(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 05 2014

a(n) ~ sqrt(6) * 24^n * (n!)^2 / (sqrt(n) * Pi^(2*n+3/2)). - Vaclav Kotesovec, May 04 2014

E.g.f.: 1/2*( 1 + Sum_{n>=0} exp((2*n+1)*x)*Product_{k=1..n} (exp((2*k-1)*x) - 1) ). Cf. A053250 and A207569. - Peter Bala, May 15 2017

Conjectural g.f.: Sum_{n >= 0} (-1)^n*Product_{k = 1..n} (1 + (-1)^k*exp(- k*t)). Cf. A158690. - Peter Bala, Jan 28 2021