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A382614 Expansion of 1/(1 - x*(1 + x)^3)^3.

%I A382614 #19 Apr 07 2025 23:41:08
%S A382614 1,3,15,55,198,681,2263,7341,23331,72928,224814,684882,2065346,
%T A382614 6173466,18310212,53935350,157904130,459755694,1332010954,3841812480,
%U A382614 11035346151,31579747613,90061069065,256028590665,725715896698,2051465107719,5784472106577,16271956316851
%N A382614 Expansion of 1/(1 - x*(1 + x)^3)^3.
%H A382614 Vincenzo Librandi, <a href="/A382614/b382614.txt">Table of n, a(n) for n = 0..400</a>
%H A382614 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (3,6,-8,-33,-24,39,108,123,84,36,9,1).
%F A382614 a(n) = Sum_{k=0..n} binomial(k+2,2) * binomial(3*k,n-k).
%F A382614 a(n) = 3*a(n-1) + 6*a(n-2) - 8*a(n-3) - 33*a(n-4) - 24*a(n-5) + 39*a(n-6) + 108*a(n-7) + 123*a(n-8) + 84*a(n-9) + 36*a(n-10) + 9*a(n-11) + a(n-12).
%F A382614 G.f.: -1/(x^4+3*x^3+3x^2+x-1)^3. - _Vincenzo Librandi_, Apr 02 2025
%t A382614 Table[Sum[Binomial[k+2,2]*Binomial[3*k,n-k],{k,0,n}],{n,0,27}] (* _Vincenzo Librandi_, Apr 02 2025 *)
%o A382614 (PARI) a(n) = sum(k=0, n, binomial(k+2, 2)*binomial(3*k, n-k));
%o A382614 (Magma) R<x> := PowerSeriesRing(Rationals(), 40); f := 1/(1 - x*(1 + x)^3)^3; seq := [ Coefficient(f, n) : n in [0..30] ]; seq; // _Vincenzo Librandi_, Apr 02 2025
%Y A382614 Column k=3 of A362125.
%Y A382614 Cf. A099234, A382613.
%Y A382614 Cf. A000217, A001628, A382406.
%Y A382614 Cf. A382615.
%K A382614 nonn,easy
%O A382614 0,2
%A A382614 _Seiichi Manyama_, Mar 31 2025