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%I #14 Sep 08 2022 08:45:25
%S 1,2,6,34,386,8706,395266,35659778,6476038146,2336999211010,
%T 1697654543745026,2450521284684021762,7120479243447937531906,
%U 41112924905741324849774594,477847273163370530909175414786
%N Row sums of triangle A118185: a(n) = Sum_{k=0..n} 4^(k*(n-k)) for n>=0.
%C Also equals column 0 of the matrix square of triangle A118185, where [A118185^2](n,k) = a(n-k)*4^(k*(n-k)) for n >= k >= 0.
%H G. C. Greubel, <a href="/A118186/b118186.txt">Table of n, a(n) for n = 0..80</a>
%F G.f.: A(x) = Sum_{n>=0} x^n/(1-4^n*x).
%F G.f.: Sum_{n>=1} a(n)*x^n/2^(n^2) = ( Sum_{n>=0} x^n/2^(n^2) )^2. - _Paul D. Hanna_, Oct 14 2009
%e A(x) = 1/(1-x) + x/(1-4x) + x^2/(1-16x) + x^3/(1-64x) + ...
%e = 1 + 2*x + 6*x^2 + 34*x^3 + 386*x^4 + 8706*x^5 + ...
%e From _Paul D. Hanna_, Oct 14 2009: (Start)
%e Another g.f.: (1 + x/2^1 + x^2/2^4 + x^3/2^9 + x^4/2^16 + ...)^2
%e = 1 + 2*x/2^1 + 6*x^2/2^4 + 34*x^3/2^9 + 386*x^4/2^16 + ... (End)
%t Table[Sum[4^(k*(n-k)), {k,0,n}], {n,0,30}] (* _G. C. Greubel_, Jun 29 2021 *)
%o (PARI) a(n)=sum(k=0, n, (4^k)^(n-k) );
%o (PARI) {a(n)=2^(n^2)*polcoeff(sum(m=0,n,x^m/2^(m^2)+x*O(x^n))^2,n)} \\ _Paul D. Hanna_, Oct 14 2009
%o (Magma) [(&+[4^(k*(n-k)): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Jun 29 2021
%o (Sage) [sum(4^(k*(n-k)) for k in (0..n)) for n in (0..30)] # _G. C. Greubel_, Jun 29 2021
%Y Cf. A118185 (triangle), A118187 (antidiagonal sums).
%K nonn
%O 0,2
%A _Paul D. Hanna_, Apr 15 2006
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