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A118187 Antidiagonal sums of triangle A118185: a(n) = Sum_{k=0..[n/2]} 4^(k*(n-2*k)) for n>=0.

%I A118187 #8 Sep 08 2022 08:45:25
%S A118187 1,1,2,5,18,81,514,5185,73730,1327361,33685506,1359217665,77311508482,
%T A118187 5567355555841,565149010231298,91215553426898945,20753150033413537794,
%U A118187 5977902509385249259521,2427296516310194305630210
%N A118187 Antidiagonal sums of triangle A118185: a(n) = Sum_{k=0..[n/2]} 4^(k*(n-2*k)) for n>=0.
%H A118187 G. C. Greubel, <a href="/A118187/b118187.txt">Table of n, a(n) for n = 0..100</a>
%F A118187 G.f.: A(x) = Sum_{n>=0} x^n/(1-4^n*x^2).
%F A118187 a(2*n) = Sum_{k=0..n} 4^(2*k*(n-k)).
%F A118187 a(2*n+1) = Sum_{k=0..n} (4^k)^(2*(n-k)+1).
%e A118187 A(x) = 1/(1-x^2) + x/(1-4*x^2) + x^2/(1-16*x^2) + x^3/(1-64*x^2) + ...
%e A118187   = 1 + x + 2*x^2 + 5*x^3 + 18*x^4 + 81*x^5 + 514*x^6 + ...
%t A118187 Table[Sum[4^(k*(n-2*k)), {k,0,Floor[n/2]}], {n,0,30}] (* _G. C. Greubel_, Jun 29 2021 *)
%o A118187 (PARI) a(n)=sum(k=0, n\2, (4^k)^(n-2*k) )
%o A118187 (Magma) [(&+[4^(k*(n-2*k)): k in [0..Floor(n/2)]]): n in [0..30]]; // _G. C. Greubel_, Jun 29 2021
%o A118187 (Sage) [sum(4^(k*(n-2*k)) for k in (0..n//2)) for n in (0..30)] # _G. C. Greubel_, Jun 29 2021
%Y A118187 Cf. A118185 (triangle), A118186 (row sums).
%K A118187 nonn
%O A118187 0,3
%A A118187 _Paul D. Hanna_, Apr 15 2006