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%I #44 Jan 28 2022 04:30:03
%S 1,9,42,140,378,882,1848,3564,6435,11011,18018,28392,43316,64260,
%T 93024,131784,183141,250173,336490,446292,584430,756470,968760,
%U 1228500,1543815,1923831,2378754,2919952,3560040,4312968,5194112,6220368,7410249,8783985,10363626
%N Partial sums of A051836.
%C If Y is a 3-subset of an n-set X then, for n >= 8, a(n-8) is the number of 8-subsets of X having at least two elements in common with Y. - _Milan Janjic_, Nov 23 2007
%C a(n) is the n-th antidiagonal sum of the convolution array A213551. - _Clark Kimberling_, Jun 17 2012
%D Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
%D Herbert John Ryser, Combinatorial Mathematics, "The Carus Mathematical Monographs", No. 14, John Wiley and Sons, 1963, pp. 1-8.
%H Vincenzo Librandi, <a href="/A051923/b051923.txt">Table of n, a(n) for n = 0..1000</a>
%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) = binomial(n+5, 5)*(n+2)/2.
%F G.f.: (1+2*x)/(1-x)^7.
%F a(n) = Sum_{k=1..n+1} k*A000217(k)*A000217(n-k+2). - _Bruno Berselli_, Sep 04 2013
%F From _Amiram Eldar_, Jan 28 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 1205/18 - 20*Pi^2/3.
%F Sum_{n>=0} (-1)^n/a(n) = 10*Pi^2/3 - 320*log(2)/3 + 755/18. (End)
%e From the third formula: a(4) = 15+60+108+120+75 = 378. - _Bruno Berselli_, Sep 04 2013
%t CoefficientList[Series[(1 + 2 x)/(1 - x)^7, {x, 0, 25}], x] (* _Harvey P. Dale_, Mar 13 2011 *)
%t Nest[Accumulate,Range[1,120,3],5] (* _Vladimir Joseph Stephan Orlovsky_, Jan 28 2012 *)
%t Table[Binomial[n + 5, 5] (n + 2) / 2, {n, 0, 35}] (* _Vincenzo Librandi_, Dec 27 2018 *)
%o (Magma) [Binomial(n+5, 5)*(n+2)/2: n in [0..40]]; // _Vincenzo Librandi_, Dec 27 2018
%Y Cf. A000217, A027801, A051836.
%Y Cf. A093560 ((3, 1) Pascal, column m=6).
%K nonn,easy
%O 0,2
%A _Barry E. Williams_, Dec 19 1999
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