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A382406 Expansion of 1/(1 - x*(1 + x)^2)^3.

%I A382406 #60 Apr 10 2025 10:46:39
%S A382406 1,3,12,37,111,315,864,2307,6027,15471,39132,97755,241606,591636,
%T A382406 1437078,3465748,8305161,19788957,46910232,110686101,260064912,
%U A382406 608684490,1419591546,3300027546,7648265728,17676484410,40747630332,93704299336,214999206831,492262973433
%N A382406 Expansion of 1/(1 - x*(1 + x)^2)^3.
%H A382406 Vincenzo Librandi, <a href="/A382406/b382406.txt">Table of n, a(n) for n = 0..500</a>
%H A382406 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (3,3,-8,-12,3,17,15,6,1).
%F A382406 a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(2*k,n-k).
%F A382406 a(n) = 3*a(n-1) + 3*a(n-2) - 8*a(n-3) - 12*a(n-4) + 3*a(n-5) + 17*a(n-6) + 15*a(n-7) + 6*a(n-8) + a(n-9).
%F A382406 G.f.: 1/(1-x-2*x^2-x^3)^3. - _Vincenzo Librandi_, Apr 10 2025
%t A382406 Table[Sum[Binomial[k+2,2]*Binomial[2*k,n-k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, Apr 10 2025 *)
%o A382406 (PARI) a(n) = sum(k=0, n, binomial(k+2, 2)*binomial(2*k, n-k));
%o A382406 (Magma) R<x> := PowerSeriesRing(Rationals(), 40); f := 1/(1 - x*(1 + x)^2)^3; seq := [ Coefficient(f, n) : n in [0..30] ]; seq; // _Vincenzo Librandi_, Apr 10 2025
%Y A382406 Cf. A002478, A362126.
%Y A382406 Cf. A000217, A001628, A382614.
%K A382406 nonn,easy
%O A382406 0,2
%A A382406 _Seiichi Manyama_, Mar 31 2025