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

%I M4935

%S 0,1,14,84,330,1001,2548,5712,11628,21945,38962,65780,106470,166257,

%T 251720,371008,534072,752913,1041846,1417780,1900514,2513049,3281916,

%U 4237520,5414500,6852105,8594586,10691604,13198654,16177505,19696656

%N Expansion of x*(1 + x)*(1 + 6*x + x^2)/(1 - x)^7.

%C Arises in enumerating paths in the plane.

%C 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 0 together with partial sums of A085461. - _Arkadiusz Wesolowski_, Aug 05 2012

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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

%H P. 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 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 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 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F a(n) = (n+1)*binomial(2n+4, 5)/12. - _Philippe Deléham_, Mar 06 2004

%e G.f. = x + 14*x^2 + 84*x^3 + 330*x^4 + 1001*x^5 + 2548*x^6 + 5712*x^7 + ...

%p series((x+7*x^2+7*x^3+x^4)/(1-x)^7,x,50);

%p 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 a[n_] := (n + 1)*Binomial[2n + 4, 5]/12;

%t Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Nov 17 2017, after _Philippe Deléham_ *)

%o (PARI) a(n) = (n+1)*binomial(2*n+4, 5)/12; \\ _Michel Marcus_, Oct 13 2016

%Y Cf. A006332.

%K nonn,easy

%O 0,3

%A _Simon Plouffe_ and _N. J. A. Sloane_

%E Edited by _N. J. A. Sloane_, Oct 20 2007

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Last modified October 22 06:02 EDT 2018. Contains 316432 sequences. (Running on oeis4.)