A172061 Expansion of (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=4.

1, 5, 22, 91, 367, 1461, 5776, 22748, 89402, 350974, 1377174, 5403193, 21201211, 83211277, 326703424, 1283211208, 5042294926, 19822108582, 77958648604, 306739666198, 1207433301046, 4754874514690, 18732340230592, 73827134976216

Offset: 0

Author

Richard Choulet, Jan 24 2010

Keywords

• easy
• nonn

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

Examples

			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.
		

Links

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

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).

Programs

Magma

k:=4; 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+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 *)

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

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

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

D-finite with recurrence: +2*(n+4)*a(n) +(-13*n-36)*a(n-1) +(15*n+16)*a(n-2) +(19*n+14)*a(n-3) +2*(2*n-1)*a(n-4)=0. - R. J. Mathar, Feb 21 2020