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A275822
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Alternating sums of the cubes of the central binomial coefficients.
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1
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1, 7, 209, 7791, 335209, 15667799, 773221225, 39651016343, 2092095886657, 112840936041343, 6193764391911873, 344853399798469695, 19429178297906958721, 1105629520934309041279, 63455683531507986958721, 3668895994183490904049279
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OFFSET
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0,2
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LINKS
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FORMULA
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a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(2*k,k)^3.
Recurrence: (n+2)^3*a(n+2)-(3*n+4)*(21*n^2+66*n+52)*a(n+1)-8*(2n+3)^3*a(n)=0.
G.f.: (4/Pi^2)*K((1-sqrt(1-64*t))/2)^2/(1+t), where K(x) is complete elliptic integral of the first kind (defined as in The Wolfram Functions Site).
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MAPLE
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L:= [seq((-1)^k*binomial(2*k, k)^3, k=0..20)]:
B:= ListTools:-PartialSums(L):
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MATHEMATICA
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Table[Sum[Binomial[2 k, k]^3 (-1)^(n - k), {k, 0, n}], {n, 0, 20}]
Table[Sum[(-1)^(n - k) (k + 1)^3 CatalanNumber[k]^3, {k, 0, n}], {n, 0, 20}] (* Jan Mangaldan, Jul 07 2020 *)
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PROG
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(Maxima) makelist(sum(binomial(2*k, k)^3*(-1)^(n-k), k, 0, n), n, 0, 12);
(PARI) a(n) = sum(k=0, n, (-1)^(n-k)*binomial(2*k, k)^3); \\ Michel Marcus, Jul 07 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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