%I #21 Oct 30 2018 20:18:09

%S 0,5,25,80,205,456,917,1708,2994,4995,7997,12364,18551,27118,38745,

%T 54248,74596,100929,134577,177080,230209,295988,376717,474996,593750,

%U 736255,906165,1107540,1344875,1623130,1947761,2324752,2760648,3262589,3838345,4496352

%N a(n) = C(n+6, 6) - n - 1.

%C In the Frey-Sellers reference this sequence is called {(n+2) over 6}_{4}, n >= 0.

%H Harry J. Smith, <a href="/A062989/b062989.txt">Table of n, a(n) for n = 0..1000</a>

%H D. D. Frey and J. A. Sellers, <a href="http://www.fq.math.ca/Scanned/39-2/frey.pdf">Generalizing Bailey's generalization of the Catalan numbers</a>, The Fibonacci Quarterly, 39 (2001) 142-148.

%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) = A062985(n+2, 6) = (n+1)*(n+2)*(n^4 + 24*n^3 + 221*n^2 + 954*n + 1800)/6!.

%F G.f.: N(5;1, x)/(1-x)^7 with N(5;1, x)= 5-10*x+10*x^2-5*x^3+x^4 = (1-(1-x)^5)/x polynomial of second row of A062986.

%F a(0)=0, a(1)=5, a(2)=25, a(3)=80, a(4)=205, a(5)=456, a(6)=917, a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7). - _Harvey P. Dale_, Aug 08 2013

%t Table[Binomial[n+6,6]-n-1,{n,0,40}] (* OR *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{0,5,25,80,205,456,917},40] (* _Harvey P. Dale_, Aug 08 2013 *)

%o (PARI) { for (n=0, 1000, write("b062989.txt", n, " ", binomial(n + 6, 6) - n - 1) ) } \\ _Harry J. Smith_, Aug 15 2009

%Y Seventh column (r=6) of FS(5) staircase array A062985.

%Y Partial sums of A062988.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Jul 12 2001

%E Simpler definition from _Zerinvary Lajos_, May 08 2006