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A099237
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a(n) = Sum_{k=0..n} binomial(n*(n-k), k).
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11
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1, 1, 3, 10, 45, 251, 1624, 11908, 97545, 880660, 8664546, 92096731, 1050304775, 12778138842, 165033693175, 2253204163256, 32401745953105, 489207829112931, 7733130368443057, 127664099576228184, 2196149923000824756
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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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a(n)^(1/n) ~ n^(n/w) * (n+1-w)^(1 - (n+1)/w) * (w-1)^(1/w - 1), where w = LambertW(exp(1)*n),
a(n)^(1/n) ~ n/log(n), but the convergence is too slow. (End)
Conjectures: a(2^k) == 1 (mod 2^k) and a(3^k) == 1 (mod 3^(k+1)); a(p^k) == 1 (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. A359842. (End)
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MAPLE
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A099237:= n-> add( binomial(n*j, n-j), j=0..n );
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MATHEMATICA
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Table[Sum[Binomial[n*(n - k), k], {k, 0, n}], {n, 0, 30}] (* Vaclav Kotesovec, Feb 19 2018 *)
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PROG
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(Sage)
def A099237(n): return sum( binomial(n*j, n-j) for j in (0..n))
(Magma)
A099237:= func< n | (&+[Binomial(n*j, n-j): j in [0..n]]) >;
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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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