A172025 Expansion of (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=3.

1, 4, 16, 62, 239, 920, 3544, 13672, 52834, 204528, 793092, 3080226, 11980667, 46662704, 181971248, 710454896, 2776717742, 10863073784, 42537035408, 166704021596, 653827252022, 2566222449104, 10079023179536, 39611016586832

Offset: 0

Author

Richard Choulet, Jan 23 2010

Keywords

• easy
• nonn

Comments

This sequence is the third 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+3 that are 1 step higher than the preceding peak. - David Scambler, Apr 22 2013

Examples

			a(4) = C(11,4) - C(10,3) + C(9,2) - C(8,1) + C(7,0) = 330 - 120 + 36 - 8 + 1 = 239.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A091526 (k=-2), A072547 (k=-1), A026641 (k=0), A014300 (k=1), A014301 (k=2), A172061 (k=4), A172062 (k=5), A172063 (k=6), A172064 (k=7), A172065 (k=8), A172066 (k=9), A172067 (k=10).

Programs

Magma

k:=3; m:=30; R:=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

Maple

a:= n-> add((-1)^(p)*binomial(2*n+3-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))^3:
a:= n-> coeff(series(gf,z,n+10),z,n):
seq(a(n), n=0..30);

Mathematica

a[n_] := Binomial[2*n+3, n+3]*Hypergeometric2F1[1, -n, -3-2*n, -1]; Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Dec 17 2013 *)

PARI

k=3; 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

Sage

k=3; ((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

Formula

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

a(n) = Sum_{p=0..n} (-1)^(p)*binomial(2*n+k-p,n-p), with k=3.

a(n) ~ 2^(2*n+4)/(3*sqrt(Pi*n)). - Vaclav Kotesovec, Apr 19 2014

Conjecture: 2*n*(n+3)*a(n) + (-7*n^2 - 17*n - 8)*a(n-1) -2*(n+2)*(2*n+1)*a(n-2) = 0. - R. J. Mathar, Feb 19 2016

a(n) = [x^n] 1/((1 - x^2)*(1 - x)^(n+3)). - Ilya Gutkovskiy, Oct 25 2017