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A005582
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a(n) = n*(n+1)*(n+2)*(n+7)/24.
(Formerly M1922)
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12
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0, 2, 9, 25, 55, 105, 182, 294, 450, 660, 935, 1287, 1729, 2275, 2940, 3740, 4692, 5814, 7125, 8645, 10395, 12397, 14674, 17250, 20150, 23400, 27027, 31059, 35525, 40455, 45880, 51832, 58344, 65450, 73185, 81585, 90687, 100529, 111150, 122590, 134890
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OFFSET
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0,2
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COMMENTS
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a(n) = number of Dyck (n+2)-paths with exactly 2 rows of peaks. A row of peaks is a maximal sequence of peaks all at the same height and 2 units apart. For example, UDUDUD ( = /\/\/\ ) contains exactly one row of peaks, as does UUUDDD, but UDUUDDUD has three and a(1)=2 counts UDUUDD, UUDDUD. - David Callan, Mar 02 2005
If X is an n-set and Y a fixed 2-subset of X then a(n-4) is equal to the number of (n-4)-subsets of X intersecting Y. - Milan Janjic, Jul 30 2007
Let I=I_n be the n X n identity matrix and P=P_n be the incidence matrix of the cycle (1,2,3,...,n). Then, for n>=7, a(n-7) is the number of (0,1) n X n matrices A<=P^(-1)+I+P having exactly two 1's in every row and column with perA=16. - Vladimir Shevelev, Apr 12 2010
a(n-1) = risefac(n, 4)/4! - risefac(n, 2)/2! is for n >= 1 also the number of independent components of a symmetric traceless tensor of rank 4 and dimension n. Here risefac is the rising factorial. - Wolfdieter Lang, Dec 10 2015
Consider the array formed by the second polygonal numbers of increasing rank:
A000217(-1-n): 0, 1, 3, 6, 10, 15, ...
A000270(-1-n): 1, 4, 9, 16, 25, 36, ...
A000326(-1-n): 2, 7, 15, 26, 40, 57, ...
A000384(-1-n): 3, 10, 21, 36, 55, 78, ...
Then the antidiagonal sums yield this sequence. - Michael Somos, Nov 23 2021
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), Table 22.7, p. 797.
Vladimir S. Shevelyov (Shevelev), Extension of the Moser class of four-line Latin rectangles, DAN Ukrainy, 3(1992),15-19. [From Vladimir Shevelev, Apr 12 2010]
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. M. Yaglom and I. M. Yaglom: Challenging Mathematical Problems with Elementary Solutions. Vol. I. Combinatorial Analysis and Probability Theory. New York: Dover Publications, Inc., 1987, p. 13, #51 (the case k=4) (First published: San Francisco: Holden-Day, Inc., 1964)
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
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a(n) = binomial(n+3, n-1) + binomial(n+2, n-1).
a(n) = binomial(n,4) + 2*binomial(n,3), n>=2. - Zerinvary Lajos, Jul 26 2006
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
G.f.: x*(2-x)/(1-x)^5. (End)
a(n+1) = A127672(8+n, n), n >= 0, with the Chebyshev C-polynomial coefficients A127672(n, k). See the Abramowitz-Stegun reference. - Wolfdieter Lang, Dec 10 2015
E.g.f.: (1/24)*x*(48 + 60*x + 16*x^2 + x^3)*exp(x). - G. C. Greubel, Jul 01 2017
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MAPLE
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[seq(binomial(n, 4)+2*binomial(n, 3), n=2..43)]; # Zerinvary Lajos, Jul 26 2006
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MATHEMATICA
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Table[n(n+1)(n+2)(n+7)/24, {n, 0, 40}] (* Harvey P. Dale, Jun 01 2012 *)
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PROG
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(PARI) concat(0, Vec(x*(2-x)/(1-x)^5 + O(x^100))) \\ Altug Alkan, Dec 10 2015
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Jun 01 2000
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STATUS
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approved
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