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A158500 Expansion of (1+sqrt(1+4x))*(1+2x)/(2*sqrt(1+4x)). 3
1, 1, 1, -4, 15, -56, 210, -792, 3003, -11440, 43758, -167960, 646646, -2496144, 9657700, -37442160, 145422675, -565722720, 2203961430, -8597496600, 33578000610, -131282408400, 513791607420, -2012616400080 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Hankel transform is A158501. Row sums of the Riordan array
((1+2x)/sqrt(1+4x), xc(-x^2))=((1-x^2)/(1+x^2),x/(1-x)^2)^{-1}, where c(x) is the g.f. of A000108.
With the proviso that the negative signs be ignored,
a(n)=the sum of the consecutive pairwise products of the terms in row(n) of Pascal's triangle. 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
Paul Barry, On a Central Transform of Integer Sequences, arXiv:2004.04577 [math.CO], 2020.
FORMULA
a(n)=C(1,n)+(-1)^n*C(2n-2,n-2).
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
MATHEMATICA
{1}~Join~Array[Binomial[1, #] + (-1)^#*Binomial[2 # - 2, # - 2] &,
24] (* Michael De Vlieger, Jul 23 2020 *)
CROSSREFS
Cf. A001791.
Sequence in context: A010905 A026030 A047038 * A001791 A047128 A087438
KEYWORD
easy,sign
AUTHOR
Paul Barry, Mar 20 2009
STATUS
approved

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Last modified June 18 15:28 EDT 2024. Contains 373481 sequences. (Running on oeis4.)