OFFSET
0,4
COMMENTS
Hankel transform is A158501.
Row sums of the Riordan array ((1+2*x)/sqrt(1+4*x), x*c(-x^2)) = ((1-x^2)/(1+x^2), x/(1-x)^2)^{-1}, where c(x) is the g.f. of A000108.
a(n) is the sum of the consecutive pairwise products of the terms in row(n) of Pascal's triangle with the proviso that the negative signs be ignored. For example, the seventh row for row(6) has the terms 1,6,15,20,15,6,1 giving a sum of 2*(1*6+6*15+15*20)=792=a(6). For row(10) the terms are 1,9,36,84,126,126,84,36,9,1 giving 2*(1*9 +9*36 +36*84 +84*126) +126*126 = 43758 = a(10). - J. M. Bergot, Jul 26 2012
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..1665
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
FORMULA
a(n) = (-1)^n*binomial(2*n-2, n-2) + binomial(1, n).
n*(n-2)*a(n) +2*(n-1)*(2*n-3)*a(n-1) = 0. - R. J. Mathar, Oct 25 2012
E.g.f.: 1 + 2*x - x*Q(0), where Q(k)= 1 + 2*x/(k+2 - (k+2)*(2*k+3)/(2*k+3 - (k+2)/Q(k+1))) (continued fraction). - Sergei N. Gladkovskii, Apr 28 2013
E.g.f.: 3/2 + x + (1/2)*(BesselI(0, 2*x) + 2*x*BesselI(0, 2*x) + 2*x*BesselI(1, 2*x))*exp(-2*x). - G. C. Greubel, Mar 17 2025
MATHEMATICA
{1}~Join~Array[Binomial[1, #] + (-1)^#*Binomial[2 #-2, #-2] &, 35] (* Michael De Vlieger, Jul 23 2020 *)
PROG
(Magma)
A158500:= func< n | n le 1 select 1 else (-1)^n*Binomial(2*n-2, n-2) >;
[A158500(n): n in [0..40]]; // G. C. Greubel, Mar 17 2025
(SageMath)
def A158500(n): return 1 if n<2 else (-1)^n*binomial(2*n-2, n-2)
print([A158500(n) for n in range(41)]) # G. C. Greubel, Mar 17 2025
CROSSREFS
KEYWORD
easy,sign
AUTHOR
Paul Barry, Mar 20 2009
STATUS
approved
