A112556 Sums of squared terms in rows of triangle A112555.
1, 2, 2, 4, 10, 32, 112, 408, 1514, 5680, 21472, 81644, 311896, 1196132, 4602236, 17757184, 68680170, 266200112, 1033703056, 4020716124, 15662273840, 61092127492, 238582873476, 932758045124, 3650336341240, 14298633670932, 56055986383412, 219931273282348, 863504076182884, 3392593262288780, 13337336618626132
Offset: 0
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
Programs
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Magma
[(1/3)*(4 - (-1/2)^n) + (n+1)*Catalan(n) - (&+[(j+1)*Catalan(j)*(-1/2)^(n-j): j in [0..n]]): n in [0..30]]; // G. C. Greubel, Jan 13 2022
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Mathematica
CoefficientList[Series[(2(1+x)/(1-x)+x/(1-4x)^(1/2))/(2+x), {x,0,30}], x] (* Harvey P. Dale, May 26 2011 *) -
PARI
{a(n)=local(x=X+X*O(X^n)); polcoeff((2*(1+x)/(1-x)+x/(1-4*x)^(1/2))/(2+x),n,X)} -
Sage
[(1/3)*(4 - (-1/2)^n) - sum( binomial(2*j, j)*(-1/2)^(n-j) for j in (0..n-1)) for n in (0..30)] # G. C. Greubel, Jan 13 2022
Formula
G.f.: ( 2*(1+x)/(1-x) + x/(1-4*x)^(1/2) )/(2+x).
a(n) ~ 2^(2*n) / (9*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 26 2016
a(n) = (1/3)*(4 - (-1/2)^n) - Sum_{j=0..n-1} binomial(2*j, j)*(-1/2)^(n-j). - G. C. Greubel, Jan 13 2022
Comments