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A006858 Expansion of x*(1 + x)*(1 + 6*x + x^2)/(1 - x)^7.
(Formerly M4935)
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 (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).

Stanley, R. P., Enumerative Combinatorics, Volume 1 (1986), p. 221, Example 4.5.18.

LINKS

Table of n, a(n) for n=0..30.

P. 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]

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

FORMULA

a(n) = (n+1)*binomial(2n+4, 5)/12. - Philippe Deléham, Mar 06 2004

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

CROSSREFS

Cf. A006332.

Sequence in context: A107935 A008451 A033276 * A027818 A054149 A273182

Adjacent sequences:  A006855 A006856 A006857 * A006859 A006860 A006861

KEYWORD

nonn,easy,changed

AUTHOR

Simon Plouffe and N. J. A. Sloane

EXTENSIONS

Edited by N. J. A. Sloane, Oct 20 2007

STATUS

approved

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Last modified November 19 01:27 EST 2017. Contains 294912 sequences.