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%I #20 Sep 08 2022 08:45:24
%S 1,1,4,6,19,33,93,175,460,910,2286,4676,11388,23842,56808,120926,
%T 283611,611065,1416625,3079635,7078263,15490553,35374519,77805481,
%U 176813809,390379483,883861033,1957097715,4418562265,9805546875,22090136885
%N Row sums of number triangle A116382.
%C Binomial transform is A116387.
%H Vincenzo Librandi, <a href="/A116383/b116383.txt">Table of n, a(n) for n = 0..200</a>
%F G.f.: (1+x-4*x^2)/(2*(1-5*x^2)*sqrt(1-4*x^2)) + (1+x)/(2*(1-5*x^2)).
%F a(n) = Sum_{k=0..n} Sum_{j=0..n} (-1)^(n-j)*C(n,j)*Sum_{i=0..j} C(j,i-k) * C(i,j-i).
%F Conjecture: n*a(n) + (n-2)*a(n-1) + (-13*n+20)*a(n-2) + (-9*n+22)*a(n-3) + 4*(14*n-45)*a(n-4) + 20*(n-3)*a(n-5) + 80*(-n+5)*a(n-6) = 0. - _R. J. Mathar_, Nov 24 2012
%F Recurrence: n*(n^2 - 4*n - 1)*a(n) = 4*(2*n-3)*a(n-1) + (9*n^3 - 40*n^2 + 3*n + 48)*a(n-2) - 20*(2*n-3)*a(n-3) - 20*(n-3)*(n^2 - 2*n - 4)*a(n-4). - _Vaclav Kotesovec_, Feb 12 2014
%F a(n) ~ (5+sqrt(5))/10 * 5^(n/2). - _Vaclav Kotesovec_, Feb 12 2014
%t CoefficientList[Series[(1+x-4*x^2)/(2*(1-5*x^2)*Sqrt[1-4*x^2])+(1+x)/(2*(1-5*x^2)), {x, 0, 40}], x] (* _Vaclav Kotesovec_, Feb 12 2014 *)
%o (PARI) my(x='x+O('x^40)); Vec((1+x-4*x^2)/(2*(1-5*x^2)*sqrt(1-4*x^2)) + (1+x)/(2*(1-5*x^2))) \\ _G. C. Greubel_, May 22 2019
%o (Magma)
%o T:= func< n,k | (&+[(-1)^(n-j)*Binomial(n,j)*(&+[Binomial(j,m-k)* Binomial(m,j-m): m in [0..j]]): j in [0..n]]) >;
%o [(&+[T(n,k): k in [0..n]]): n in [0..40]]; // _G. C. Greubel_, May 22 2019
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1+x-4*x^2)/(2*(1-5*x^2)*Sqrt(1-4*x^2)) + (1+x)/(2*(1-5*x^2)) )); // _G. C. Greubel_, May 22 2019
%o (Sage) ((1+x-4*x^2)/(2*(1-5*x^2)*sqrt(1-4*x^2)) + (1+x)/(2*(1-5*x^2))).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, May 22 2019
%o (GAP) List([0..40], n-> Sum([0..n], k-> Sum([0..n], j-> (-1)^(n-j)* Binomial(n,j)*Sum([0..j], m-> Binomial(j,m-k)*Binomial(m,j-m) )))) # _G. C. Greubel_, May 22 2019
%K easy,nonn
%O 0,3
%A _Paul Barry_, Feb 12 2006
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