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A104970 Sum of squares of terms in even-indexed rows of triangle A104967.

%I A104970 #15 Sep 08 2022 08:45:17
%S A104970 1,6,18,92,298,1444,4852,22840,78490,362580,1265564,5767688,20366596,
%T A104970 91866984,327351336,1464522864,5257011066,23361650484,84371466636,
%U A104970 372831130344,1353477992556,5952169844664,21704580414936,95051752387344
%N A104970 Sum of squares of terms in even-indexed rows of triangle A104967.
%C A104970 Sum of squares of terms in odd-indexed rows of triangle A104967 equals twice this sequence.
%H A104970 Vincenzo Librandi, <a href="/A104970/b104970.txt">Table of n, a(n) for n = 0..200</a>
%F A104970 G.f. A(x) satisfies: 2*(1+12*x)*A(x) - (1-16*x^2)*deriv(A(x), x) + 4 = 0.
%F A104970 a(n) = 2^(2*n-1)*(2 + Sum_{k=0..n-1} (-1)^k*binomial(2*k+1,k+1)/2^(2*k)). - _Vaclav Kotesovec_, Oct 28 2012
%t A104970 Flatten[{1,Table[2^(2*n-1)*(2+Sum[(-1)^k*Binomial[2*k+1,k+1]/2^(2*k),{k,0,n-1}]),{n,1,20}]}] (* _Vaclav Kotesovec_, Oct 28 2012 *)
%o A104970 (PARI) {a(n)=local(X=x+x*O(x^(2*n))); sum(k=0,2*n,polcoeff(polcoeff((1-2*X)/(1-X-X*y*(1-2*X)),2*n,x),k,y)^2)}
%o A104970 (Magma)
%o A104970 A104970:= func< n | n eq 0 select 1 else  4^n + (&+[(-1)^j*2^(2*n-2*j-1)*Binomial(2*j+1,j+1): j in [0..n-1]]) >;
%o A104970 [A104970(n): n in [0..40]]; // _G. C. Greubel_, Jun 09 2021
%o A104970 (Sage)
%o A104970 @cached_function
%o A104970 def A104967(n,k): return sum( (-2)^j*binomial(k+1,j)*binomial(n-j,k) for j in (0..n-k))
%o A104970 def A104970(n): return sum((A104967(2*n,k))^2 for k in (0..2*n))
%o A104970 [A104970(n) for n in (0..50)] # _G. C. Greubel_, Jun 09 2021
%Y A104970 Cf. A104967, A104968, A104969.
%K A104970 nonn
%O A104970 0,2
%A A104970 _Paul D. Hanna_, Mar 30 2005