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A113409
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A transform of the central binomial coefficients A001405.
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2
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1, 1, 2, 3, 6, 11, 21, 39, 74, 141, 271, 521, 1004, 1939, 3756, 7291, 14176, 27599, 53805, 105031, 205268, 401573, 786328, 1541037, 3022528, 5932657, 11652617, 22901865, 45037432, 88616807, 174454943, 343606183, 677074350, 1334744305
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OFFSET
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0,3
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COMMENTS
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LINKS
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FORMULA
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G.f.: (1-xc(x^2))/(1-x^2-x^4c(x^4)), where c(x) is the g.f. of A000108.
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*C(k, floor(k/2)).
a(n) = Sum_{k=0..n} C((n+k)/2, k)*C(floor((n-k)/2), floor((n-k)/4)).
Conjecture: (n+2)*a(n)-2*(n+1)*a(n-1) +(n-4)*a(n-2) +2*a(n-3) +4*(2-n)*a(n-4)=0. - R. J. Mathar, Nov 07 2012
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MATHEMATICA
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Table[Sum[Binomial[n - k, k]*Binomial[k, Floor[k/2]], {k, 0, Floor[n/2]}], {n, 0, 50}] (* G. C. Greubel, Mar 09 2017 *)
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PROG
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(PARI) for(n=0, 25, print1(sum(k=0, floor(n/2), binomial(n-k, k)*binomial(k, floor(k/2))), ", ")) \\ G. C. Greubel, Mar 09 2017
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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