OFFSET
0,2
COMMENTS
Sixth row of number array A082137. C(n,5) has e.g.f. (x^5/5!)exp(x). The transform averages the binomial and inverse binomial transforms.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..400
Index entries for linear recurrences with constant coefficients, signature (12,-60,160,-240,192,-64).
FORMULA
Equals 2 * A080952.
a(n) = (2^(n-1) + 0^n/2)*C(n+5, n).
a(n) = Sum_{j=0..n} C(n+5, j+5)*C(j+5, 5)*(1+(-1)^j)/2.
G.f.: (1 -6*x +30*x^2 -80*x^3 +120*x^4 -96*x^5 +32*x^6)/(1-2*x)^6.
E.g.f.: x^5*exp(x)*cosh(x)/5! (preceded by 5 zeros).
a(n) = ceiling(binomial(n+5,5)*2^(n-1)). - Zerinvary Lajos, Nov 01 2006
From Amiram Eldar, Jan 07 2022: (Start)
Sum_{n>=0} 1/a(n) = 20*log(2) - 38/3.
Sum_{n>=0} (-1)^n/a(n) = 1620*log(3/2) - 656. (End)
EXAMPLE
a(0) = (2^(-1) + 0^0/2)*binomial(5,0) = 2*(1/2) = 1 (use 0^0 = 1).
MAPLE
[seq (ceil(binomial(n+5, 5)*2^(n-1)), n=0..23)]; # Zerinvary Lajos, Nov 01 2006
MATHEMATICA
Drop[With[{nmax = 56}, CoefficientList[Series[x^5*Exp[x]*Cosh[x]/5!, {x, 0, nmax}], x]*Range[0, nmax]!], 5] (* or *) Join[{1}, Table[2^(n-1)* Binomial[n+5, n], {n, 1, 30}] (* G. C. Greubel, Feb 05 2018 *)
PROG
(Magma) [(Ceiling(Binomial(n+5, 5)*2^(n-1))) : n in [0..30]]; // Vincenzo Librandi, Sep 22 2011
(PARI) my(x='x+O('x^30)); Vec(serlaplace(x^5*exp(x)*cosh(x)/5!)) \\ G. C. Greubel, Feb 05 2018
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Apr 06 2003
STATUS
approved