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A028724 a(n) = (1/2)*floor(n/2)*floor((n-1)/2)*floor((n-2)/2). 6
0, 0, 0, 0, 1, 2, 6, 9, 18, 24, 40, 50, 75, 90, 126, 147, 196, 224, 288, 324, 405, 450, 550, 605, 726, 792, 936, 1014, 1183, 1274, 1470, 1575, 1800, 1920, 2176, 2312, 2601, 2754, 3078, 3249, 3610, 3800, 4200, 4410, 4851, 5082, 5566, 5819, 6348, 6624, 7200 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

Number of symmetric Dyck paths of semilength n and having four peaks. E.g., a(5)=2 because we have UU*DU*DU*DU*DD and U*DUU*DU*DDU*D, where U=(1,1), D=(1,-1) and * indicates peaks. - Emeric Deutsch, Jan 12 2004

Starting with "1" = triangle A171608 * the triangular numbers. - Gary W. Adamson, Dec 12 2009

REFERENCES

P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 185, Article 433.

LINKS

Iain Fox, Table of n, a(n) for n = 0..10000

P. Barry, On sequences with {-1, 0, 1} Hankel transforms, arXiv preprint arXiv:1205.2565 [math.CO], 2012. - From N. J. A. Sloane, Oct 18 2012

Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).

FORMULA

G.f.: x^4(1+x+x^2)/(x-1)^4/(x+1)^3. - Ralf Stephan, Jun 22 2003

Number of tuples [x, y, z, w] of integers such that n = x + y, x >= max(y, z), min(y, z) >= w >= 2. - Michael Somos, Jan 27 2008

Euler transform of length 3 sequence [2, 3, -1]. - Michael Somos, Jan 27 2008

a(3-n) = -a(n). - Michael Somos, Jan 27 2008

a(n) = (11*n - 3 - 9*n^2 + 2*n^3 + (-1)^n*(3 - 3*n + n^2))/32. - Benedict W. J. Irwin, Sep 27 2016

a(n) = Sum_{i=1..floor((n-1)/2)} i * ( floor((n-1)/2) mod (n-i-1) ). - Wesley Ivan Hurt, Nov 17 2017

EXAMPLE

a(7) = 9 since the 9 tuples [x, y, z, w] in {[4, 3, 2, 2] [4, 3, 3, 2] [4, 3, 3, 3] [4, 3, 4, 2] [4, 3, 4, 3] [5, 2, 2, 2] [5, 2, 3, 2] [5, 2, 4, 2] [5, 2, 5, 2]} are all the solutions of 7 = x + y, x >= max(y, z), min(y, z) >= w >= 2.

MAPLE

A028724:=n->(1/2)*floor(n/2)*floor((n-1)/2)*floor((n-2)/2);

seq(A028724(k), k=0..100); # Wesley Ivan Hurt, Nov 01 2013

MATHEMATICA

Table[(1/2)*Floor[n/2]*Floor[(n-1)/2]*Floor[(n-2)/2], {n, 0, 100}] (* Wesley Ivan Hurt, Nov 01 2013 *)

Table[(11n-3-9n^2+2n^3+(-1)^n(3-3n+n^2))/32, {n, 0, 30}] (* Benedict W. J. Irwin, Sep 27 2016 *)

CoefficientList[Series[x^4 (1 + x + x^2)/(x - 1)^4/(x + 1)^3, {x, 0, 50}], x] (* Michael De Vlieger, Sep 27 2016 *)

PROG

(PARI) {a(n) = (n\2) * ((n-1)\2) * (n\2-1) / 2} /* Michael Somos, Jan 27 2008 */

(PARI) {a(n) = if( n<0, n=-1-n; -1, n-=4; 1) * polcoeff( (1 - x^3) / (1 - x)^2 / (1 - x^2)^3 + x*O(x^n), n)} /* Michael Somos, Jan 27 2008 */

(PARI) first(n) = Vec(x^4*(1+x+x^2)/(x-1)^4/(x+1)^3 + O(x^(n)), -n) \\ Iain Fox, Nov 18 2017

CROSSREFS

Cf. A171608.

Sequence in context: A032471 A156222 A002886 * A222048 A156190 A285446

Adjacent sequences:  A028721 A028722 A028723 * A028725 A028726 A028727

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified October 17 19:44 EDT 2019. Contains 328128 sequences. (Running on oeis4.)