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%I #10 Sep 08 2022 08:45:31
%S 1,3,8,17,34,64,117,209,368,641,1108,1904,3257,5551,9432,15985,27030,
%T 45616,76845,129245,217056,364033,609768,1020192,1705009,2846619,
%U 4748072,7912529,13174858,21919456,36440613,60538409,100503632,166744961,276476092,458151440
%N Row sums of triangle A131252.
%H Andrew Howroyd, <a href="/A131253/b131253.txt">Table of n, a(n) for n = 0..500</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4, -4, -2, 4, 0, -1)
%F From _Andrew Howroyd_, Aug 09 2018: (Start)
%F a(n) = Sum_{k=0..n} (k+1)*(Sum_{i=0..k} binomial(n-k, k-i)).
%F a(n) = 4*a(n-1) - 4*a(n-2) - 2*a(n-3) + 4*a(n-4) - a(n-6).
%F G.f.: (1 - x - x^3)/((1 - x)^2*(1 - x - x^2)^2).
%F (End)
%e a(3) = 17 = sum of row 3 terms of A131252: (7 + 6 + 3 + 1).
%t LinearRecurrence[{4, -4, -2, 4, 0, -1}, {1, 3, 8, 17, 34, 64}, 40] (* _Vincenzo Librandi_, Aug 10 2018 *)
%o (PARI) Vec((1 - x - x^3)/((1 - x)^2*(1 - x - x^2)^2) + O(x^40)) \\ _Andrew Howroyd_, Aug 09 2018
%o (PARI) a(n)={sum(k=0, n, (k+1)*sum(i=0, k, binomial(n-k, k-i)))} \\ _Andrew Howroyd_, Aug 09 2018
%o (Magma) I:=[1,3,8,17,34,64]; [n le 6 select I[n] else 4*Self(n-1)- 4*Self(n-2)-2*Self(n-3)+4*Self(n-4)-Self(n-6): n in [1..40]]; // _Vincenzo Librandi_, Aug 10 2018
%Y Row sums of A131252.
%K nonn
%O 0,2
%A _Gary W. Adamson_, Jun 23 2007
%E Terms a(10) and beyond from _Andrew Howroyd_, Aug 09 2018