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A118192 Antidiagonal sums of triangle A118190: a(n) = Sum_{k=0..floor(n/2)} 5^(k*(n-2*k)) for n>=0.

%I A118192 #8 Sep 08 2022 08:45:25
%S A118192 1,1,2,6,27,151,1252,18876,421877,11797501,489062502,36867190626,
%T A118192 4119892578127,576049853531251,119400024902343752,
%U A118192 45003894807128984376,25145828723919677734377,17579646409034759521875001
%N A118192 Antidiagonal sums of triangle A118190: a(n) = Sum_{k=0..floor(n/2)} 5^(k*(n-2*k)) for n>=0.
%H A118192 G. C. Greubel, <a href="/A118192/b118192.txt">Table of n, a(n) for n = 0..100</a>
%F A118192 G.f.: A(x) = Sum_{n>=0} x^n/(1-5^n*x^2).
%F A118192 a(2*n) = Sum_{k=0..n} 5^(2*k*(n-k)).
%F A118192 a(2*n+1) = Sum_{k=0..n} 5^(k*(2*(n-k)+1)).
%e A118192 A(x) = 1/(1-x^2) + x/(1-5*x^2) + x^2/(1-25*x^2) + x^3/(1-125*x^2) + ...
%e A118192   = 1 + x + 2*x^2 + 6*x^3 + 27*x^4 + 151*x^5 + ...
%t A118192 Table[Sum[5^(k*(n-2*k)), {k,0,Floor[n/2]}], {n,0,30}] (* _G. C. Greubel_, Jun 29 2021 *)
%o A118192 (PARI) a(n)=sum(k=0, n\2, (5^k)^(n-2*k) )
%o A118192 (Magma) [(&+[5^(k*(n-2*k)): k in [0..Floor(n/2)]]): n in [0..30]]; // _G. C. Greubel_, Jun 29 2021
%o A118192 (Sage) [sum(5^(k*(n-2*k)) for k in (0..n//2)) for n in (0..30)] # _G. C. Greubel_, Jun 29 2021
%Y A118192 Cf. A118190 (triangle), A118191 (row sums).
%K A118192 nonn
%O A118192 0,3
%A A118192 _Paul D. Hanna_, Apr 15 2006