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A118182 Antidiagonal sums of triangle A118180: a(n) = Sum_{k=0..[n/2]} (3^k)^(n-2*k) for n>=0.

%I A118182 #8 Sep 08 2022 08:45:25
%S A118182 1,1,2,4,11,37,164,1000,8021,81001,1076006,19683244,473632031,
%T A118182 14349084877,571833704648,31381448626000,2265367321680041,
%U A118182 205893684435186001,24615565942378859210,4052605390737766057684
%N A118182 Antidiagonal sums of triangle A118180: a(n) = Sum_{k=0..[n/2]} (3^k)^(n-2*k) for n>=0.
%H A118182 G. C. Greubel, <a href="/A118182/b118182.txt">Table of n, a(n) for n = 0..125</a>
%F A118182 G.f.: A(x) = Sum_{n>=0} x^n/(1-3^n*x^2).
%F A118182 a(2*n) = Sum_{k=0..n} (3^k)^(2*(n-k)).
%F A118182 a(2*n+1) = Sum_{k=0..n} (3^k)^(2*(n-k) +1).
%e A118182 A(x) = 1/(1-x^2) + x/(1-3x^2) + x^2/(1-9x^2) + x^3/(1-27x^2) +...
%e A118182 = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 37*x^5 + 164*x^6 + 1000*x^7 +...
%t A118182 Table[Sum[3^(k*(n-2*k)), {k,0,Floor[n/2]}], {n,0,30}] (* _G. C. Greubel_, Jun 29 2021 *)
%o A118182 (PARI) a(n)=sum(k=0, n\2, (3^k)^(n-2*k) );
%o A118182 (Magma) [(&+[3^(k*(n-2*k)): k in [0..Floor(n/2)]]): n in [0..30]]; // _G. C. Greubel_, Jun 29 2021
%o A118182 (Sage) [sum(3^(k*(n-2*k)) for k in (0..n//2)) for n in (0..30)] # _G. C. Greubel_, Jun 29 2021
%Y A118182 Cf. A118180 (triangle), A118181 (row sums), A118183, A118184.
%K A118182 nonn
%O A118182 0,3
%A A118182 _Paul D. Hanna_, Apr 15 2006