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A006858 Expansion of g.f. x*(1 + x)*(1 + 6*x + x^2)/(1 - x)^7.
(Formerly M4935)
12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Arises in enumerating paths in the plane.
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
0 together with partial sums of A085461. - Arkadiusz Wesolowski, Aug 05 2012
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
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]
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
EXAMPLE
G.f. = x + 14*x^2 + 84*x^3 + 330*x^4 + 1001*x^5 + 2548*x^6 + 5712*x^7 + ...
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)];
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 *)
PROG
(PARI) a(n) = (n+1)*binomial(2*n+4, 5)/12; \\ Michel Marcus, Oct 13 2016
(Sage) [(n+1)*binomial(2*n+4, 5)/12 for n in (0..30)] # G. C. Greubel, Dec 14 2021
CROSSREFS
Sequence in context: A107935 A008451 A033276 * A027818 A054149 A273182
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane, Oct 20 2007
STATUS
approved

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)