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A006857 a(n) = binomial(n+5,5) * binomial(n+5,4)/(n+5).

%I A006857 M4977 #80 Mar 12 2025 08:06:51
%S A006857 1,15,105,490,1764,5292,13860,32670,70785,143143,273273,496860,866320,
%T A006857 1456560,2372112,3755844,5799465,8756055,12954865,18818646,26883780,
%U A006857 37823500,52474500,71867250,97260345,130179231,172459665,226296280,294296640,379541184
%N A006857 a(n) = binomial(n+5,5) * binomial(n+5,4)/(n+5).
%C A006857 Number of permutations of n+5 that avoid the pattern 132 and have exactly 4 descents.
%C A006857 Kekulé numbers for certain benzenoids. - _Emeric Deutsch_, Nov 18 2005
%C A006857 Partial sums of A114242. - _Peter Bala_, Sep 21 2007
%C A006857 Dimensions of certain Lie algebra (see reference for precise definition).
%D A006857 S. J. Cyvin and I. Gutman, Kekulé structures in benzenoid hydrocarbons, Lecture Notes in Chemistry, No. 46, Springer, New York, 1988 (pp. 167-169, Table 10.5/II/1).
%D A006857 S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 239.
%D A006857 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006857 T. D. Noe, <a href="/A006857/b006857.txt">Table of n, a(n) for n = 0..1000</a>
%H A006857 Brandy Amanda Barnette, <a href="https://digitalcommons.wku.edu/theses/1484">Counting Convex Sets on Products of Totally Ordered Sets</a>, Masters Theses & Specialist Projects, Paper 1484, 2015.
%H A006857 G. Kreweras, <a href="http://www.numdam.org/item?id=BURO_1967__10__23_0">Traitement simultané du "Problème de Young" et du "Problème de Simon Newcomb"</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle. Institut de Statistique, Université de Paris, 10 (1967), 23-31.
%H A006857 G. Kreweras, <a href="/A006542/a006542_1.pdf">Traitement simultané du "Problème de Young" et du "Problème de Simon Newcomb"</a>, Cahiers du Bureau Universitaire de Recherche Opérationnelle. Institut de Statistique, Université de Paris, 10 (1967), 23-31. [Annotated scanned copy]
%H A006857 J. M. Landsberg and L. Manivel, <a href="http://dx.doi.org/10.1016/j.aim.2005.02.001">The sextonions and E7 1/2</a>, Adv. Math. 201 (2006), 143-179. [Th. 7.3, case a=4]
%H A006857 Feihu Liu, Guoce Xin, and Chen Zhang, <a href="https://arxiv.org/abs/2412.18744">Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS</a>, arXiv:2412.18744 [math.CO], 2024. See p. 25.
%H A006857 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).
%F A006857 From - _Vladeta Jovovic_, Jan 29 2003: (Start)
%F A006857 a(n) = (n+4)!*(n+5)!/(2880*n!*(n+1)!).
%F A006857 E.g.f.: 1/2880*(2880 + 40320*x + 109440*x^2 + 105120*x^3 + 45000*x^4 + 9504*x^5 + 1016*x^6 + 52*x^7 + x^8)*exp(x). (End)
%F A006857 From _Mike Zabrocki_, Aug 26 2004: (Start)
%F A006857 a(n) = C(n+5,8) + 6*C(n+6,8) + 6*C(n+7,8) + C(n+8,8).
%F A006857 a(n) = C(n+4,4)*C(n+5,4)/5.
%F A006857 O.g.f.: (1 + 6*x + 6*x^2 + x^3)/(1-x)^9. (End)
%F A006857 From _Wolfdieter Lang_, Nov 13 2007: (Start)
%F A006857 a(n) = A001263(n+5,5).
%F A006857 Numerator polynomial of the g.f is the fourth row polynomial of the Narayana triangle. (End)
%F A006857 a(n)= C(n+4,4)^2 - C(n+4,3)*C(n+4,5). - _Gary Detlefs_, Dec 05 2011
%F A006857 a(n) = Product_{i=1..4} A002378(n+i)/A002378(i). - _Bruno Berselli_, Sep 01 2016
%F A006857 From _Amiram Eldar_, Oct 19 2020: (Start)
%F A006857 Sum_{n>=0} 1/a(n) = 25 * (79 - 8*Pi^2).
%F A006857 Sum_{n>=0} (-1)^n/a(n) = 595/3 - 20*Pi^2. (End)
%p A006857 a:=n->(n+1)*(n+2)^2*(n+3)^2*(n+4)^2*(n+5)/2880: seq(a(n),n=0..38); # _Emeric Deutsch_, Nov 18 2005
%t A006857 Table[Binomial[n+5,5] * Binomial[n+5,4]/(n+5), {n, 0, 50}] (* _T. D. Noe_, May 29 2012 *)
%t A006857 LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,15,105,490,1764,5292,13860,32670,70785},40] (* _Harvey P. Dale_, Oct 19 2024 *)
%o A006857 (PARI) a(n) = binomial(n+5,5) * binomial(n+5,4)/(n+5) \\ _Charles R Greathouse IV_, Jun 11 2015
%o A006857 (PARI) Vec((1+6*x+6*x^2+x^3)/(1-x)^9 + O(x^99)) \\ _Altug Alkan_, Sep 01 2016
%o A006857 (Magma)
%o A006857 A006857:= func< n | Binomial(n+4,3)*Binomial(n+5,5)/4 >;
%o A006857 [A006857(n): n in [0..40]]; // _G. C. Greubel_, Mar 12 2025
%o A006857 (SageMath)
%o A006857 def A006857(n): return binomial(n+4,3)*binomial(n+5,5)//4
%o A006857 print([A006857(n) for n in range(41)]) # _G. C. Greubel_, Mar 12 2025
%Y A006857 The expression binomial(m+n-1,n)^2-binomial(m+n,n+1)*binomial(m+n-2,n-1) for the values m = 2 through 14 produces the sequences A000012, A000217, A002415, A006542, A006857, A108679, A134288, A134289, A134290, A134291, A140925, A140935, A169937.
%Y A006857 5th column of the table of Narayana numbers A001263.
%Y A006857 Cf. A002378, A114242.
%K A006857 nonn,easy
%O A006857 0,2
%A A006857 _Simon Plouffe_
%E A006857 More terms from _Vladeta Jovovic_, Jan 29 2003
%E A006857 Better description from _Mike Zabrocki_, Aug 26 2004
%E A006857 New definition from _N. J. A. Sloane_, Aug 28 2010
%E A006857 Zabrocki formulas offset corrected by _Gary Detlefs_, Dec 05 2011