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URL: https://oeis.org/A104496

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A104496
Expansion of 2*(2*x+1)/((x+1)*(sqrt(4*x+1)+1)).
4
1, 0, 0, -1, 5, -19, 67, -232, 804, -2806, 9878, -35072, 125512, -452388, 1641028, -5986993, 21954973, -80884423, 299233543, -1111219333, 4140813373, -15478839553, 58028869153, -218123355523, 821908275547, -3104046382351, 11747506651599, -44546351423299, 169227201341651
OFFSET
0,5
COMMENTS
Previous name was: Row sums of triangle A104495. A104495 equals the matrix inverse of triangle A099602, where row n of A099602 equals the inverse Binomial transform of column n of the triangle of trinomial coefficients (A027907).
Absolute row sums of triangle A104495 forms A014137 (partial sums of Catalan numbers).
LINKS
FORMULA
G.f.: A(x) = (1 + 2*x)/(1+x)/(1+x - x^2*Catalan(-x)^2), where Catalan(x)=(1-(1-4*x)^(1/2))/(2*x) (cf. A000108).
a(n) ~ (-1)^n * 2^(2*n+1) / (3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Mar 06 2014
D-finite with recurrence: (n+1)*a(n) +(7*n-3)*a(n-1) +2*(7*n-12)*a(n-2) +4*(2*n-5)*a(n-3)=0. - R. J. Mathar, Jan 23 2020
MAPLE
gf := (2*(2*x+1))/((x+1)*(sqrt(4*x+1)+1)): ser := series(gf, x, 30):
seq(coeff(ser, x, n), n=0..28); # Peter Luschny, Apr 25 2016
MATHEMATICA
CoefficientList[Series[(1+2*x)/(1+x)/(1+x - (1-(1+4*x)^(1/2))^2/4), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 06 2014 *)
PROG
(PARI) {a(n)=local(X=x+x*O(x^n)); polcoeff( (1+2*X)/(1+X)/(1+X-(1-(1+4*X)^(1/2))^2/4), n, x)}
(Python)
from itertools import accumulate
def A104496_list(size):
if size < 1: return []
L, accu = [1], [1]
for n in range(size-1):
accu = list(accumulate(accu + [-accu[0]]))
L.append(-(-1)**n*accu[-1])
return L
print(A104496_list(29)) # Peter Luschny, Apr 25 2016
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Paul D. Hanna, Mar 11 2005
EXTENSIONS
New name using the g.f. of the author by Peter Luschny, Apr 25 2016
STATUS
approved