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%I #25 Jul 14 2015 12:56:34
%S 1,82,2107,29332,274357,1930726,10948735,52357960,217994860,808970960,
%T 2723733524,8436372248,24304813148,65712993248,167965846148,
%U 408373664744,949291256585,2119095737210,4559798912835,9488531918460,19148848609485,37571357310510
%N a(n) = sum_{i=1..n} C(7+i,8)^2.
%C In general we have the following formula : a(n) = Sum_{i=1..n}C(e-1+i,e)^2 = C(n+e-1,e+1)*Fe(n)/C(2*e+1). We have the following definition : Fe(n) = Sum_{i=1..n}(-1)^(e+i)*C(e+i,i)*C(n+e,i), and Fe(1) = C(2*e+1,e). [This needs clarification, _Joerg Arndt_, May 04 2014]
%H Vincenzo Librandi, <a href="/A234253/b234253.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (18, -153, 816, -3060, 8568, -18564, 31824, -43758, 48620, -43758, 31824, -18564, 8568, -3060, 816, -153, 18, -1).
%F a(n) = sum_{i=1..n)C(7+i,8)^2 = C(n+8,9)*F8(n)/C(17,8) ; F8(n)= Sum_{i=1..n}(-1)^(8+i)*C(8+i,i)*C(n+8,i) = C(8,0)*C(n+8,0) - C(9,1)*C(n+8,1) + C(10,2)*C(n+8,2) - C(11,3)*C(n+8,3) + C(12,4)*C(n+8,4) - C(13,5)*C(n+8,5) + C(14,6)*C(n+8,6) - C(15,7)*C(n+8,7) + C(16,8)*Cn+8,8). We have the following values for F8(n) : F8(0)=1, F8(1)=24310, F8(2)=199342, F8(3)=931294, .... [This needs clarification, _Joerg Arndt_, May 04 2014]
%F G.f.: x*(x^8 +64*x^7 +784*x^6 +3136*x^5 +4900*x^4 +3136*x^3 +784*x^2 +64*x +1) / (x-1)^18. - _Colin Barker_, May 02 2014
%e For n=3, Sum_{i=1..3)C(7+i,8)^2 = C(11,9)*F8(3)/C(17,8) = 55*931294/24310 = 2107. [This needs clarification, _Joerg Arndt_, May 04 2014]
%p A234253:=n->add(binomial(7+i,8)^2, i=1..n); seq(A234253(n), n=1..30); # _Wesley Ivan Hurt_, Dec 23 2013
%t Table[Sum[Binomial[7 + i, 8]^2, {i, n}], {n, 30}] (* _Wesley Ivan Hurt_, Dec 23 2013 *)
%t CoefficientList[Series[(x^8 + 64 x^7 + 784 x^6 + 3136 x^5 + 4900 x^4 + 3136 x^3 + 784 x^2 + 64 x + 1)/(x - 1)^18, {x, 0, 40}], x] (* _Vincenzo Librandi_, May 06 2014 *)
%o (PARI) Vec(x*(x^8 +64*x^7 +784*x^6 +3136*x^5 +4900*x^4 +3136*x^3 +784*x^2 +64*x +1)/(x-1)^18 + O(x^100)) \\ _Colin Barker_, May 02 2014
%Y Cf. A087127, A086023, A086025, A086027, A086029.
%K nonn,easy
%O 1,2
%A _Yahia Kahloune_, Dec 22 2013
%E One term corrected and more terms from _Colin Barker_, May 02 2014
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