A006858 Expansion of g.f. x*(1 + x)*(1 + 6*x + x^2)/(1 - x)^7.
0, 1, 14, 84, 330, 1001, 2548, 5712, 11628, 21945, 38962, 65780, 106470, 166257, 251720, 371008, 534072, 752913, 1041846, 1417780, 1900514, 2513049, 3281916, 4237520, 5414500, 6852105, 8594586, 10691604, 13198654, 16177505, 19696656, 23831808, 28666352, 34291873
Offset: 0
Examples
G.f. = x + 14*x^2 + 84*x^3 + 330*x^4 + 1001*x^5 + 2548*x^6 + 5712*x^7 + ...
References
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Richard P. Stanley, Enumerative Combinatorics, Volume 1, 1986, p. 221, Example 4.5.18.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Paolo Aluffi, Degrees of projections of rank loci, 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]."]
- G. Kreweras and H. Niederhausen, Solution of an enumerative problem connected with lattice paths, European J. Combin., 2 (1981), 55-60.
- J. M. Landsberg and L. Manivel, The sextonions and E7 1/2, Adv. Math. 201 (2006), 143-179. [Th. 7.2(ii), case a=1]
- Feihu Liu, Guoce Xin, and Chen Zhang, Ehrhart Polynomials of Order Polytopes: Interpreting Combinatorial Sequences on the OEIS, arXiv:2412.18744 [math.CO], 2024. See pp. 9, 24.
- Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).
Programs
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Maple
series((x+7*x^2+7*x^3+x^4)/(1-x)^7,x,50); 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)];
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Mathematica
a[n_]:= (n+1)*Binomial[2n+4, 5]/12; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 17 2017, after Philippe Deléham *) -
PARI
a(n) = (n+1)*binomial(2*n+4, 5)/12; \\ Michel Marcus, Oct 13 2016
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Sage
[(n+1)*binomial(2*n+4, 5)/12 for n in (0..30)] # G. C. Greubel, Dec 14 2021
Formula
a(n) = (n+1)*binomial(2*n+4, 5)/12. - Philippe Deléham, Mar 06 2004
a(n) = a(-2-n) for all n in Z. - Michael Somos, Jun 27 2023
From Amiram Eldar, Jul 09 2023: (Start)
Sum_{n>=1} 1/a(n) = 30*Pi^2 - 295.
Sum_{n>=1} (-1)^(n+1)/a(n) = -15*Pi^2 + 240*Pi - 605. (End)
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
Extensions
Edited by N. J. A. Sloane, Oct 20 2007
Comments