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A110145
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a(n) = Sum_{k=0..n} C(n,k)^2*mod(k,2).
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4
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0, 1, 4, 10, 32, 126, 472, 1716, 6400, 24310, 92504, 352716, 1351616, 5200300, 20060016, 77558760, 300533760, 1166803110, 4537591960, 17672631900, 68923172032, 269128937220, 1052049834576, 4116715363800, 16123800489472, 63205303218876, 247959271674352
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OFFSET
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0,3
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COMMENTS
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Number of n-element subsets of [2n] having an odd sum. - Alois P. Heinz, Feb 06 2017
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} C(n, k)^2*(1-(-1)^k)/2.
a(n) = C(2n-1, n-1)(1-(-1)^n)/2+(C(2n, n)/2-(-1)^(n/2)*C(n, floor(n/2))/2)(1+(-1)^n)/2.
a(n) = (binomial(2*n, n) - binomial(n, n/2)*cos(Pi*n/2))/2 = n^2 * hypergeom([1/2-n/2, 1/2-n/2, 1-n/2, 1-n/2], [1, 3/2, 3/2], 1). - Vladimir Reshetnikov, Oct 04 2016
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MATHEMATICA
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Table[Sum[Binomial[n, k]^2 Mod[k, 2], {k, 0, n}], {n, 0, 30}] (* Harvey P. Dale, Feb 21 2013 *)
Table[(Binomial[2 n, n] - Binomial[n, n/2] Cos[Pi n/2])/2, {n, 0, 30}] (* Vladimir Reshetnikov, Oct 04 2016 *)
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PROG
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(PARI) a(n) = sum(k=0, n, binomial(n, k)^2*(k % 2)); \\ Michel Marcus, Oct 05 2016
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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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