A358050 -id:A358050 - cp's OEIS frontend

A358145 a(n) = Sum_{k=0..n} binomial(nk,k) binomial(n*(n-k),n-k).

1, 2, 16, 258, 6184, 195660, 7674144, 358788696, 19464910000, 1201543131276, 83134800597280, 6371436086078382, 535715287899894216, 49025879014213908144, 4850781409411286177248, 515964243167132532702480, 58710263012322890445554400

Offset: 0

Seiichi Manyama, Oct 31 2022

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..337

Main diagonal of A358050.

a(n) = sum(k=0, n, binomial(n*k, k)*binomial(n*(n-k), n-k));

a(n) = sum(k=0, n, (n-1)^(n-k)*binomial(n^2+1, k));

a(n) = sum(k=0, n, n^(n-k)*binomial((n-1)*n+k, k));

a(n) = Sum_{k=0..n} (n-1)^(n-k) * binomial(n^2+1,k).
a(n) = Sum_{k=0..n} n^(n-k) * binomial((n-1)*n+k,k).
a(n) ~ exp(n - 1/2) * n^n / 2. - Vaclav Kotesovec, Nov 01 2022

A358146 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(k*j,j).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 4, 9, 4, 1, 1, 5, 19, 29, 5, 1, 1, 6, 33, 103, 99, 6, 1, 1, 7, 51, 253, 598, 351, 7, 1, 1, 8, 73, 506, 2073, 3601, 1275, 8, 1, 1, 9, 99, 889, 5351, 17577, 22165, 4707, 9, 1, 1, 10, 129, 1429, 11515, 58481, 152173, 138445, 17577, 10, 1

Offset: 0

Seiichi Manyama, Oct 31 2022

			Square array begins:
  1, 1,   1,    1,     1,     1, ...
  1, 2,   3,    4,     5,     6, ...
  1, 3,   9,   19,    33,    51, ...
  1, 4,  29,  103,   253,   506, ...
  1, 5,  99,  598,  2073,  5351, ...
  1, 6, 351, 3601, 17577, 58481, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0-5 give: A000012, A001477(n+1), A006134, A188675, A225612, A225615.
Main diagonal gives A226391.
Cf. A358050.

T(n, k) = sum(j=0, n, binomial(k*j, j));

cp's OEIS Frontend

A358145 a(n) = Sum_{k=0..n} binomial(nk,k) binomial(n*(n-k),n-k).

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A358146 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(k*j,j).

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cp's OEIS Frontend

A358145 a(n) = Sum_{k=0..n} binomial(n*k,k) * binomial(n*(n-k),n-k).

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Author

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PARI

PARI

PARI

Formula

A358146 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} binomial(k*j,j).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A358145 a(n) = Sum_{k=0..n} binomial(nk,k) binomial(n*(n-k),n-k).