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A006858 Expansion of g.f. x*(1 + x)*(1 + 6*x + x^2)/(1 - x)^7.

%I A006858 M4935 #71 Jan 07 2025 12:55:16
%S A006858 0,1,14,84,330,1001,2548,5712,11628,21945,38962,65780,106470,166257,
%T A006858 251720,371008,534072,752913,1041846,1417780,1900514,2513049,3281916,
%U A006858 4237520,5414500,6852105,8594586,10691604,13198654,16177505,19696656,23831808,28666352,34291873
%N A006858 Expansion of g.f. x*(1 + x)*(1 + 6*x + x^2)/(1 - x)^7.
%C A006858 Arises in enumerating paths in the plane.
%C A006858 a(n+1) is the determinant of the n X n Hankel matrix whose first row is the Catalan numbers C_n (A000108) beginning at C_4 = 14. Example (n=3): det[{{14, 42, 132}, {42, 132, 429}, {132, 429, 1430}}] = 330. - _David Callan_, Mar 30 2007
%C A006858 0 together with partial sums of A085461. - _Arkadiusz Wesolowski_, Aug 05 2012
%D A006858 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A006858 Richard P. Stanley, Enumerative Combinatorics, Volume 1, 1986, p. 221, Example 4.5.18.
%H A006858 G. C. Greubel, <a href="/A006858/b006858.txt">Table of n, a(n) for n = 0..1000</a>
%H A006858 Paolo Aluffi, <a href="https://arxiv.org/abs/1408.1702">Degrees of projections of rank loci</a>, arXiv:1408.1702 [math.AG], 2014. ["After compiling the results of many explicit computations, we noticed that many of the numbers d_{n,r,S} appear in the existing literature in contexts far removed from the enumerative geometry of rank conditions; we owe this surprising (to us) observation to perusal of [Slo14]."]
%H A006858 G. Kreweras and H. Niederhausen, <a href="http://dx.doi.org/10.1016/S0195-6698(81)80020-0">Solution of an enumerative problem connected with lattice paths</a>, European J. Combin., 2 (1981), 55-60.
%H A006858 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. 7.2(ii), case a=1]
%H A006858 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 pp. 9, 24.
%H A006858 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).
%F A006858 a(n) = (n+1)*binomial(2*n+4, 5)/12. - _Philippe Deléham_, Mar 06 2004
%F A006858 a(n) = a(-2-n) for all n in Z. - _Michael Somos_, Jun 27 2023
%F A006858 From _Amiram Eldar_, Jul 09 2023: (Start)
%F A006858 Sum_{n>=1} 1/a(n) = 30*Pi^2 - 295.
%F A006858 Sum_{n>=1} (-1)^(n+1)/a(n) = -15*Pi^2 + 240*Pi - 605. (End)
%F A006858 E.g.f.: exp(x)*x*(180 + 1080*x + 1350*x^2 + 555*x^3 + 84*x^4 + 4*x^5)/180. - _Stefano Spezia_, Dec 09 2023
%e A006858 G.f. = x + 14*x^2 + 84*x^3 + 330*x^4 + 1001*x^5 + 2548*x^6 + 5712*x^7 + ...
%p A006858 series((x+7*x^2+7*x^3+x^4)/(1-x)^7,x,50);
%p A006858 b:=binomial; t72b:= proc(a,k) ((a+k+1)/(a+1)) * b(k+2*a+1,k)*b(k+3*a/2+1,k)/(b(k+a/2,k)); end; [seq(t72b(1,k),k=0..40)];
%t A006858 a[n_]:= (n+1)*Binomial[2n+4, 5]/12;
%t A006858 Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Nov 17 2017, after _Philippe Deléham_ *)
%o A006858 (PARI) a(n) = (n+1)*binomial(2*n+4, 5)/12; \\ _Michel Marcus_, Oct 13 2016
%o A006858 (Sage) [(n+1)*binomial(2*n+4, 5)/12 for n in (0..30)] # _G. C. Greubel_, Dec 14 2021
%Y A006858 Cf. A000108, A006332, A085461.
%K A006858 nonn,easy
%O A006858 0,3
%A A006858 _Simon Plouffe_ and _N. J. A. Sloane_
%E A006858 Edited by _N. J. A. Sloane_, Oct 20 2007