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A028724
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(1/2)*floor(n/2)*floor((n-1)/2)*floor((n-2)/2).
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5
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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; internal format)
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OFFSET
| 0,6
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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 (deutsch(AT)duke.poly.edu), Jan 12 2004
Starting with "1" = triangle A171608 * the triangular numbers. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 12 2009]
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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.
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).
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FORMULA
| G.f. x^4(1+x+x^2)/(x-1)^4/(x+1)^3. - Ralf Stephan (ralf(AT)ark.in-berlin.de), 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
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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.
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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 */
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CROSSREFS
| Cf. A171608.
Sequence in context: A032471 A156222 A002886 * A156190 A076738 A082290
Adjacent sequences: A028721 A028722 A028723 * A028725 A028726 A028727
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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