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A359842
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a(n) = Sum_{k=0..n} binomial(n*k,n+k).
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2
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1, 0, 1, 90, 13690, 3443275, 1308315371, 701623884514, 505274768721332, 470638793249281593, 550707386335951810915, 790898932162231992184327, 1367864138835420575101044139, 2804370191530797723173615407860, 6725366044028696102055907486691290
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OFFSET
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0,4
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LINKS
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FORMULA
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a(n) ~ binomial(n^2,2*n).
a(n) ~ exp(2*n-2) * n^(2*n - 1/2) / (sqrt(Pi) * 2^(2*n+1)).
Conjectures: a(2^k) == 0 (mod 2^(k-1)) and a(3^k) == 0 (mod 3^(k+2)) for k >= 2; a(p^k) == 0 (mod p^(k+1)) for all primes p >= 5.
Let m be a positive integer. Similar recurrences may hold for the sequence whose n-th term is given by Sum_{k = 0..n} binomial(m*n*k, n+k). Cf. A099237. (End)
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MAPLE
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a := proc (n) option remember; add(binomial(n*k, n+k), k = 0..n) end:
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MATHEMATICA
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Table[Sum[Binomial[n*k, n+k], {k, 0, n}], {n, 0, 20}]
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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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