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A129369 Expansion of 1/sqrt(1-4*x) - x/sqrt(1-4*x^2). 5
1, 1, 6, 18, 70, 246, 924, 3412, 12870, 48550, 184756, 705180, 2704156, 10399676, 40116600, 155114088, 601080390, 2333593350, 9075135300, 35345215180, 137846528820, 538257689684, 2104098963720, 8233430022168, 32247603683100 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Partial sums are A129368.
LINKS
FORMULA
a(n) = binomial(2*n, n) - (1/2)*(1-(-1)^n)*binomial(n-1, (n-1)/2).
From G. C. Greubel, Jan 31 2024: (Start)
a(2*n) = A001448(n).
a(2*n+1) = (n+1)*( 2*A000108(2*n+1) - A000108(n) ).
a(2*n+1) = 2*A002458(n) - A000984(n).
(n+1)^3*(n+2)*(n+3)*a(n+3) = 2*(n+1)*(n+2)^3*(2*n+1)*a(n+2) + 4*(n+1)^4*(n+3)*a(n+1) - 8*n*(n+2)^3*(2*n+1)*a(n), with a(0)=a(1) = 1, a(2) = 6. (End)
MATHEMATICA
CoefficientList[Series[1/Sqrt[1-4x]-x/Sqrt[1-4x^2], {x, 0, 30}], x] (* Harvey P. Dale, Feb 02 2012 *)
PROG
(Magma) B:=Binomial; [B(2*n, n) - (n mod 2)*B(n-1, Floor((n-1)/2)): n in [0..60]]; // G. C. Greubel, Jan 31 2024
(SageMath) [binomial(2*n, n) - (n%2)*binomial(n-1, (n-1)//2) for n in range(61)] # G. C. Greubel, Jan 31 2024
CROSSREFS
Sequence in context: A200151 A000623 A304941 * A095853 A027266 A242278
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Apr 11 2007
STATUS
approved

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Last modified July 15 16:36 EDT 2024. Contains 374333 sequences. (Running on oeis4.)