|
%I #12 Sep 08 2022 08:45:24
%S 1,0,1,3,4,7,16,29,52,102,194,361,685,1301,2452,4633,8771,16577,31327,
%T 59241,112004,211724,400285,756786,1430710,2704817,5113647,9667590,
%U 18277014,34553692,65325542,123501151,233485250,441415867,834519021
%N Expansion of 1/(1-x^2*(1+x)^3).
%C Diagonal sums of number triangle A116089.
%H G. C. Greubel, <a href="/A116090/b116090.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (0, 1, 3, 3, 1).
%F a(n) = a(n-2) + 3*a(n-3) + 3*a(n-4) + a(n-5).
%F a(n) = Sum_{k=0..floor(n/2)} C(3*k, n-2*k).
%F a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*C(4*k,n-k)/C(4*k,k).
%t CoefficientList[Series[1/(1-x^2(1+x)^3),{x,0,40}],x] (* or *) LinearRecurrence[{0,1,3,3,1},{1,0,1,3,4},40] (* _Harvey P. Dale_, Apr 28 2014 *)
%o (PARI) {a(n) = sum(k=0, floor(n/2), binomial(3*k, n-2*k))}; \\ _G. C. Greubel_, May 09 2019
%o (Magma) [(&+[Binomial(3*k, n-2*k): k in [0..Floor(n/2)]]): n in [0..40]]; // _G. C. Greubel_, May 09 2019
%o (Sage) [sum(binomial(3*k, n-2*k) for k in (0..floor(n/2))) for n in (0..40)] # _G. C. Greubel_, May 09 2019
%K easy,nonn
%O 0,4
%A _Paul Barry_, Feb 04 2006
|
