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

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

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

G. C. Greubel, Table of n, a(n) for n = 0..1000

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(2*n+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
(Sage) [(n+1)*binomial(2*n+4, 5)/12 for n in (0..30)] # G. C. Greubel, Dec 14 2021

Crossrefs

Cf. A006332, A085461.

Sequence in context: A107935 A008451 A033276 * A027818 A054149 A273182

Adjacent sequences: A006855 A006856 A006857 * A006859 A006860 A006861

Keyword

nonn,easy

Author

Simon Plouffe and N. J. A. Sloane

Extensions

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

Status

approved