login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A172025 Expansion of (2/(3*sqrt(1-4*z)-1+4*z))*((1-sqrt(1-4*z))/(2*z))^k with k=3. 12
1, 4, 16, 62, 239, 920, 3544, 13672, 52834, 204528, 793092, 3080226, 11980667, 46662704, 181971248, 710454896, 2776717742, 10863073784, 42537035408, 166704021596, 653827252022, 2566222449104, 10079023179536, 39611016586832 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

LINKS

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

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

EXAMPLE

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

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-Fran├žois Alcover, Dec 17 2013 *)

PROG

(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

(MAGMA) k:=3; 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=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

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).

Sequence in context: A113438 A268429 A195339 * A171278 A227438 A206839

Adjacent sequences:  A172022 A172023 A172024 * A172026 A172027 A172028

KEYWORD

easy,nonn

AUTHOR

Richard Choulet, Jan 23 2010

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 7 20:33 EST 2019. Contains 329849 sequences. (Running on oeis4.)