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A062989 a(n) = C(n+6, 6) - n - 1.

%I A062989 #25 Jan 05 2025 19:51:36
%S A062989 0,5,25,80,205,456,917,1708,2994,4995,7997,12364,18551,27118,38745,
%T A062989 54248,74596,100929,134577,177080,230209,295988,376717,474996,593750,
%U A062989 736255,906165,1107540,1344875,1623130,1947761,2324752,2760648,3262589,3838345,4496352
%N A062989 a(n) = C(n+6, 6) - n - 1.
%C A062989 In the Frey-Sellers reference this sequence is called {(n+2) over 6}_{4}, n >= 0.
%H A062989 Harry J. Smith, <a href="/A062989/b062989.txt">Table of n, a(n) for n = 0..1000</a>
%H A062989 D. D. Frey and J. A. Sellers, <a href="https://web.archive.org/web/2024*/https://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 A062989 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7, -21, 35, -35, 21, -7, 1).
%F A062989 a(n) = A062985(n+2, 6) = (n+1)*(n+2)*(n^4 + 24*n^3 + 221*n^2 + 954*n + 1800)/6!.
%F A062989 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 A062989 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
%F A062989 D-finite with recurrence -n*a(n) +(n+6)*a(n-1) +5*n=0. - _R. J. Mathar_, Nov 22 2024
%t A062989 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 A062989 (PARI) { for (n=0, 1000, write("b062989.txt", n, " ", binomial(n + 6, 6) - n - 1) ) } \\ _Harry J. Smith_, Aug 15 2009
%Y A062989 Seventh column (r=6) of FS(5) staircase array A062985.
%Y A062989 Partial sums of A062988.
%K A062989 nonn,easy
%O A062989 0,2
%A A062989 _Wolfdieter Lang_, Jul 12 2001
%E A062989 Simpler definition from _Zerinvary Lajos_, May 08 2006