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A119363
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a(n) = Sum_{k=0..n} C(n,3k)^2.
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7
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1, 1, 1, 2, 17, 101, 402, 1275, 3921, 14114, 58601, 243695, 950578, 3537847, 13166791, 50514102, 198627921, 782913717, 3054480306, 11824753551, 45823049817, 178682390994, 700285942731, 2747647985241, 10767833451954, 42164261091351, 165225573240651
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OFFSET
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0,4
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COMMENTS
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LINKS
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FORMULA
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Recurrence: (n-2)*(n-1)*n*(637*n^6 - 11466*n^5 + 84364*n^4 - 324394*n^3 + 686227*n^2 - 755060*n + 336132)*a(n) = 3*(n-2)*(n-1)*(1274*n^7 - 23569*n^6 + 180194*n^5 - 733383*n^4 + 1699606*n^3 - 2208294*n^2 + 1449504*n - 351000)*a(n-1) - 3*(n-2)*(3185*n^8 - 63700*n^7 + 539028*n^6 - 2512118*n^5 + 7020469*n^4 - 11971242*n^3 + 12050010*n^2 - 6446736*n + 1362744)*a(n-2) + (14014*n^9 - 315315*n^8 + 3072678*n^7 - 16986046*n^6 + 58535088*n^5 - 129861691*n^4 + 184326992*n^3 - 159830656*n^2 + 75517728*n - 14313456)*a(n-3) + 3*(n-3)*(3185*n^8 - 63700*n^7 + 538391*n^6 - 2501394*n^5 + 6946794*n^4 - 11707256*n^3 + 11530544*n^2 - 5915328*n + 1142208)*a(n-4) + 18*(n-4)*(n-3)*(2*n - 9)*(637*n^6 - 7644*n^5 + 36589*n^4 - 88858*n^3 + 114124*n^2 - 71840*n + 16440)*a(n-5).
a(n) ~ 4^n / (3*sqrt(Pi*n)). (End)
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MATHEMATICA
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Table[Sum[Binomial[n, 3k]^2, {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Mar 12 2019 *)
Table[HypergeometricPFQ[{1/3 - n/3, 1/3 - n/3, 2/3 - n/3, 2/3 - n/3, -n/3, -n/3}, {1/3, 1/3, 2/3, 2/3, 1}, 1], {n, 0, 30}] (* Vaclav Kotesovec, Mar 12 2019 *)
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CROSSREFS
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Central coefficients of number triangle A119335.
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KEYWORD
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easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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