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%I M1922 #106 Apr 13 2022 13:25:17
%S 0,2,9,25,55,105,182,294,450,660,935,1287,1729,2275,2940,3740,4692,
%T 5814,7125,8645,10395,12397,14674,17250,20150,23400,27027,31059,35525,
%U 40455,45880,51832,58344,65450,73185,81585,90687,100529,111150,122590,134890
%N a(n) = n*(n+1)*(n+2)*(n+7)/24.
%C 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
%C 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
%C 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
%C Row 2 of the convolution array A213550. - _Clark Kimberling_, Jun 20 2012
%C 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
%C Consider the array formed by the second polygonal numbers of increasing rank:
%C A000217(-1-n): 0, 1, 3, 6, 10, 15, ...
%C A000270(-1-n): 1, 4, 9, 16, 25, 36, ...
%C A000326(-1-n): 2, 7, 15, 26, 40, 57, ...
%C A000384(-1-n): 3, 10, 21, 36, 55, 78, ...
%C Then the antidiagonal sums yield this sequence. - _Michael Somos_, Nov 23 2021
%D 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.
%D 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]
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D 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)
%H Vincenzo Librandi, <a href="/A005582/b005582.txt">Table of n, a(n) for n = 0..10000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Richard K. Guy, <a href="/A005581/a005581_1.pdf">Letter to N. J. A. Sloane, Feb 1988</a>
%H F. T. Howard and Curtis Cooper, <a href="http://www.fq.math.ca/Papers1/49-3/HowardCooper.pdf">Some identities for r-Fibonacci numbers</a>, Fibonacci Quart. 49 (2011), no. 3, 231-243.
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H P. A. MacMahon, <a href="http://plms.oxfordjournals.org/content/s2-23/1/290.extract">Properties of prime numbers deduced from the calculus of symmetric functions</a>, Proc. London Math. Soc., 23 (1923), 290-316. = Coll. Papers, II, pp. 354-380. [See p. 301]
%H Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992
%H C. Rossiter, <a href="https://web.archive.org/web/20130515133733/http://noticingnumbers.net/300SeriesCube.htm">Depictions, Explorations and Formulas of the Euler/Pascal Cube</a>.
%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = binomial(n+3, n-1) + binomial(n+2, n-1).
%F a(n) = binomial(n,4) + 2*binomial(n,3), n>=2. - _Zerinvary Lajos_, Jul 26 2006
%F From _Colin Barker_, Jan 28 2012: (Start)
%F a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F G.f.: x*(2-x)/(1-x)^5. (End)
%F a(n) = Sum_{k=1..n} ( Sum_{i=1..k} i(n-k+2) ). - _Wesley Ivan Hurt_, Sep 26 2013
%F 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
%F E.g.f.: (1/24)*x*(48 + 60*x + 16*x^2 + x^3)*exp(x). - _G. C. Greubel_, Jul 01 2017
%F Sum_{n>=1} 1/a(n) = 853/1225. - _Amiram Eldar_, Jan 02 2021
%F a(n) = A005587(-7-n) for all n in Z. - _Michael Somos_, Nov 23 2021
%p [seq(binomial(n,4)+2*binomial(n,3), n=2..43)]; # _Zerinvary Lajos_, Jul 26 2006
%p seq((n+4)*binomial(n,4)/n, n=3..43); # _Zerinvary Lajos_, Feb 28 2007
%p A005582:=(-2+z)/(z-1)**5; # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t Table[n(n+1)(n+2)(n+7)/24,{n,0,40}] (* _Harvey P. Dale_, Jun 01 2012 *)
%o (PARI) concat(0, Vec(x*(2-x)/(1-x)^5 + O(x^100))) \\ _Altug Alkan_, Dec 10 2015
%Y Partial sums of A005581.
%Y Cf. A000211, A052928, A128209, A176222.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from Larry Reeves (larryr(AT)acm.org), Jun 01 2000
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