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A172061 Expansion of (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=4. 13
1, 5, 22, 91, 367, 1461, 5776, 22748, 89402, 350974, 1377174, 5403193, 21201211, 83211277, 326703424, 1283211208, 5042294926, 19822108582, 77958648604, 306739666198, 1207433301046, 4754874514690, 18732340230592, 73827134976216 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This sequence is the 4th diagonal below the main diagonal (which itself is A026641) in the array which grows with "Pascal rule" given here by rows: 1,0,1,0,1,0,1,0,1,0,1,0,1,0, 1,1,1,1,1,1,1,1,1,1,1,1,1,1, 1,1,2,2,3,3,4,4,5,5,6,6,7,7, 1,2,4,6,9,12,16,20,25,30, 1,3,7,13,22,34,50,70,95. The MAPLE programs give the first diagonals of this array.

Apparently the number of peaks in all Dyck paths of semilength n+4 that are 2 steps higher than the preceding peak. - David Scambler, Apr 22 2013

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

FORMULA

G.f.: (2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k with k=4.

a(n) = Sum_{p=0..n} (-1)^(p)*binomial(2*n+k-p, n-p), with k=4.

a(n) ~ 2^(2*n+5)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 19 2014

Conjecture: 2*n*(n+4)*(3*n+5)*a(n) +(-21*n^3-107*n^2-176*n-96)*a(n-1) -2*(3*n+8)*(2*n+3)*(n+1)*a(n-2)=0. - R. J. Mathar, Feb 19 2016

EXAMPLE

a(4) = C(12,4)-C(11,3)+C(10,2)-C(9,1)+C(8,0)=55*9-55*3+45-9+1=367.

MAPLE

a:= n-> add((-1)^(p)*binomial(2*n+4-p, n-p), p=0..n):

seq(a(n), n=0..30);

# second Maple program:

gf:= (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^4:

a:= n-> coeff(series(gf, z, n+10), z, n):

seq(a(n), n=0..30);

MATHEMATICA

CoefficientList[Series[(2/(3*Sqrt[1-4*x]-1+4*x))*((1-Sqrt[1-4*x])/(2*x))^4, {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 19 2014 *)

PROG

(PARI) k=4; my(x='x+O('x^30)); Vec((2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k) \\ G. C. Greubel, Feb 16 2019

(MAGMA) k:=4; m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (2/(3*Sqrt(1-4*x)-1+4*x))*((1-Sqrt(1-4*x))/(2*x))^k )); // G. C. Greubel, Feb 16 2019

(Sage) k=4; ((2/(3*sqrt(1-4*x)-1+4*x))*((1-sqrt(1-4*x))/(2*x))^k).series(x, 20).coefficients(x, sparse=False) # G. C. Greubel, Feb 16 2019

CROSSREFS

Cf. A091526 (k=-2), A072547 (k=-1), A026641 (k=0), A014300 (k=1), A014301 (k=2), A172025 (k=3), A172062 (k=5), A172063 (k=6), A172064 (k=7), A172065 (k=8), A172066 (k=9), A172067 (k=10).

Sequence in context: A050185 A216597 A085812 * A211973 A053297 A071715

Adjacent sequences:  A172058 A172059 A172060 * A172062 A172063 A172064

KEYWORD

easy,nonn

AUTHOR

Richard Choulet, Jan 24 2010

STATUS

approved

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Last modified December 15 11:43 EST 2019. Contains 329999 sequences. (Running on oeis4.)