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A349846
Expansion of -(1 - 8*x) / sqrt(1 - 4*x).
3
-1, 6, 10, 28, 90, 308, 1092, 3960, 14586, 54340, 204204, 772616, 2939300, 11232648, 43088200, 165815280, 639859770, 2475036900, 9593714460, 37255818600, 144915581580, 564514356120, 2201964031800, 8599360982160, 33619842137700, 131570223027048, 515366318553912
OFFSET
0,2
COMMENTS
Sum_{n>=0} (-a(n)/(-4)^n) is the Cauchy product of Sum_{n>=0} (-A349844(n)/(-8)^n) with itself.
FORMULA
For n > 0, a(n) = 8*binomial(2*(n-1),n-1) - binomial(2*n,n) = binomial(2*(n-1),n-1) * (4 + 2/n).
a(n) ~ 4^n * (1/sqrt(Pi*n)).
EXAMPLE
a(1) = binomial(0,0) * (4 + 2/1) = 6;
a(2) = binomial(2,1) * (4 + 2/2) = 10;
a(3) = binomial(4,2) * (4 + 2/3) = 28;
a(4) = binomial(6,3) * (4 + 2/4) = 90.
PROG
(PARI) a(n) = if(n, binomial(2*(n-1), n-1) * (4 + 2/n), -1)
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Jianing Song, Dec 01 2021
STATUS
approved