A104970 Sum of squares of terms in even-indexed rows of triangle A104967.
1, 6, 18, 92, 298, 1444, 4852, 22840, 78490, 362580, 1265564, 5767688, 20366596, 91866984, 327351336, 1464522864, 5257011066, 23361650484, 84371466636, 372831130344, 1353477992556, 5952169844664, 21704580414936, 95051752387344
Offset: 0
Keywords
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
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Magma
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]]) >; [A104970(n): n in [0..40]]; // G. C. Greubel, Jun 09 2021
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Mathematica
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 *) -
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)} -
Sage
@cached_function def A104967(n,k): return sum( (-2)^j*binomial(k+1,j)*binomial(n-j,k) for j in (0..n-k)) def A104970(n): return sum((A104967(2*n,k))^2 for k in (0..2*n)) [A104970(n) for n in (0..50)] # G. C. Greubel, Jun 09 2021
Formula
G.f. A(x) satisfies: 2*(1+12*x)*A(x) - (1-16*x^2)*deriv(A(x), x) + 4 = 0.
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
Comments