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A062990 Eighth column (r=7) of FS(5) staircase array A062985.

%I A062990 #28 Jan 05 2025 19:51:36
%S A062990 5,30,110,315,771,1688,3396,6390,11385,19382,31746,50297,77415,116160,
%T A062990 170408,245004,345933,480510,657590,887799,1183787,1560504,2035500,
%U A062990 2629250,3365505,4271670,5379210,6724085,8347215
%N A062990 Eighth column (r=7) of FS(5) staircase array A062985.
%C A062990 In the Frey-Sellers reference this sequence is called {(n+2) over 7}_{4}, n >= 0.
%H A062990 Harry J. Smith, <a href="/A062990/b062990.txt">Table of n, a(n) for n = 0..1000</a>
%H A062990 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 A062990 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%F A062990 a(n) = A062985(n+2, 7) = (n+1)*(n+2)*(n+3)*(n^4 + 29*n^3 + 326*n^2 + 1744*n + 4200)/7!.
%F A062990 G.f.: N(5;1, x)/(1-x)^8 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 A062990 a(n) = binomial(n+7,n) - binomial(n+2,n). - _Zerinvary Lajos_, Jun 23 2006
%p A062990 [seq((binomial(n+7,n)-binomial(n+2,n)),n=1..29)]; # _Zerinvary Lajos_, Jun 23 2006
%t A062990 Table[Binomial[n+7,n]-Binomial[n+2,n],{n,30}] (* or *) LinearRecurrence[ {8,-28,56,-70,56,-28,8,-1},{5,30,110,315,771,1688,3396,6390},30] (* _Harvey P. Dale_, Jun 09 2016 *)
%o A062990 (PARI) { for (n=0, 1000, m=n + 1; a=binomial(m + 7, m) - binomial(m + 2, m); write("b062990.txt", n, " ", a) ) } \\ _Harry J. Smith_, Aug 15 2009
%Y A062990 Partial sums of A062989.
%K A062990 nonn,easy
%O A062990 0,1
%A A062990 _Wolfdieter Lang_, Jul 12 2001
%E A062990 More terms from _Zerinvary Lajos_, Jun 23 2006