A359842 a(n) = Sum_{k=0..n} binomial(n*k,n+k).

1, 0, 1, 90, 13690, 3443275, 1308315371, 701623884514, 505274768721332, 470638793249281593, 550707386335951810915, 790898932162231992184327, 1367864138835420575101044139, 2804370191530797723173615407860, 6725366044028696102055907486691290

Offset

0,4

Links

Seiichi Manyama, Table of n, a(n) for n = 0..205

Formula

a(n) ~ binomial(n^2,2*n).

a(n) ~ exp(2*n-2) * n^(2*n - 1/2) / (sqrt(Pi) * 2^(2*n+1)).

From Peter Bala, Jan 19 2023: (Start)

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)

Maple

a := proc (n) option remember; add(binomial(n*k, n+k), k = 0..n) end:
seq(a(n), n = 0..20); # Peter Bala, Jan 16 2023

Mathematica

Table[Sum[Binomial[n*k, n+k], {k, 0, n}], {n, 0, 20}]

Crossrefs

Cf. A096131, A099237, A226391.

Sequence in context: A036257 A276352 A203779 * A295594 A200503 A199229

Adjacent sequences: A359839 A359840 A359841 * A359843 A359844 A359845

Keyword

nonn

Author

Vaclav Kotesovec, Jan 15 2023

Status

approved