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A062966 a(n) = C(3+n, n) + C(4+n, n) + C(5+n, n) + C(6+n, n).

%I A062966 #24 Jan 05 2025 19:51:36
%S A062966 4,22,74,195,441,896,1680,2958,4950,7942,12298,18473,27027,38640,
%T A062966 54128,74460,100776,134406,176890,229999,295757,376464,474720,593450,
%U A062966 735930,905814,1107162,1344469,1622695,1947296
%N A062966 a(n) = C(3+n, n) + C(4+n, n) + C(5+n, n) + C(6+n, n).
%C A062966 In the Frey-Sellers reference this sequence is called {(n+2) over 6}_{3}, n >= 0.
%H A062966 Harry J. Smith, <a href="/A062966/b062966.txt">Table of n, a(n) for n = 0..1000</a>
%H A062966 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 A062966 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (7, -21, 35, -35, 21, -7, 1).
%F A062966 a(n) = A062750(n+2, 6) = (n+10)*(n+3)*(n+2)*(n+1)*(n^2+11*n+48)/6!.
%F A062966 G.f.: = (x-2)*(x^2-2*x+2)/(x-1)^7 = N(4;1, x)/(1-x)^7 with N(4;1, x) = 4-6*x+4*x^2-x^3, polynomial of second row of A062751.
%F A062966 a(0)=4, a(1)=22, a(2)=74, a(3)=195, a(4)=441, a(5)=896, a(6)=1680, 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_, May 02 2012
%t A062966 Table[Sum[Binomial[i+n,n],{i,3,6}],{n,0,30}] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{4,22,74,195,441,896,1680},30] (* _Harvey P. Dale_, May 02 2012 *)
%o A062966 (PARI) { for (n=0, 1000, a=binomial(3 + n, n) + binomial(4 + n, n) + binomial(5 + n, n) + binomial(6 + n, n); write("b062966.txt", n, " ", a) ) } \\ _Harry J. Smith_, Aug 14 2009
%Y A062966 Seventh column (r=6) of FS(4) staircase array A062750.
%Y A062966 Partial sums of A027659.
%K A062966 nonn,easy
%O A062966 0,1
%A A062966 _Wolfdieter Lang_, Jul 12 2001
%E A062966 Better description from _Zerinvary Lajos_, Dec 02 2005