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A104496 Expansion of 2*(2*x+1)/((x+1)*(sqrt(4*x+1)+1)). 3
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 (list; graph; refs; listen; history; text; internal format)
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

G. C. Greubel, Table of n, a(n) for n = 0..1000

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

Cf. A104495, A099602, A027907, A000108.

Sequence in context: A163872 A035344 A114277 * A001435 A092492 A070857

Adjacent sequences:  A104493 A104494 A104495 * A104497 A104498 A104499

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

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Last modified July 7 14:13 EDT 2020. Contains 335495 sequences. (Running on oeis4.)