A336522 -id:A336522 - cp's OEIS frontend

A336537 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n^2+k+1,n)/(n^2+k+1).

1, 2, 10, 134, 3298, 122762, 6208970, 399606286, 31331798914, 2902190030354, 310441644900682, 37685712807847062, 5120833751373831138, 770270980249401539482, 127088854993223378639498, 22824507222500649365932062, 4432992797251355031727570434, 925899965014326913556521154594

Offset: 0

Seiichi Manyama, Jul 25 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..311

Main diagonal of A336534.

Cf. A336522, A336577.

a[0] = 1; a[n_] := Sum[2^k * Binomial[n, k] * Binomial[n^2, k - 1], {k, 1, n}] / n; Array[a, 18, 0] (* Amiram Eldar, Jul 25 2020 *)

{a(n) = sum(k=0, n, binomial(n, k) * binomial(n^2+k+1, n)/(n^2+k+1))}

{a(n) = if(n==0, 1, sum(k=1, n, 2^k*binomial(n, k) * binomial(n^2, k-1)/n))}

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

a(n) = (1/n) * Sum_{k=1..n} 2^k * binomial(n,k) * binomial(n^2,k-1) for n > 0.
a(n) = (1/(n^2+1)) * Sum_{k=0..n} binomial(n^2+1,k) * binomial((n+1)*n-k,n-k).
a(n) ~ 2^(n - 1/2) * exp(n) * n^(n - 5/2) / sqrt(Pi). - Vaclav Kotesovec, Jul 31 2021
a(n) = 2*hypergeom([1-n, -n^2], [2], 2) for n > 0. - Stefano Spezia, Aug 09 2025

A336521 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is the coefficient of x^(k*n) in expansion of ( (1 + x)/(1 - x) )^n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 8, 1, 1, 2, 16, 38, 1, 1, 2, 24, 146, 192, 1, 1, 2, 32, 326, 1408, 1002, 1, 1, 2, 40, 578, 4672, 14002, 5336, 1, 1, 2, 48, 902, 11008, 69002, 142000, 28814, 1, 1, 2, 56, 1298, 21440, 216002, 1038984, 1459810, 157184, 1, 1, 2, 64, 1766, 36992, 525002, 4320608, 15856206, 15158272, 864146, 1

Offset: 0

Seiichi Manyama, Jul 24 2020

			Square array begins:
  1,    1,     1,     1,      1,      1, ...
  1,    2,     2,     2,      2,      2, ...
  1,    8,    16,    24,     32,     40, ...
  1,   38,   146,   326,    578,    902, ...
  1,  192,  1408,  4672,  11008,  21440, ...
  1, 1002, 14002, 69002, 216002, 525002, ...

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

Column k=0-3 give A000012, A123164, A103885, A333715.
Main diagonal gives A336522.
Cf. A122542, A266213, A336534.

T[n_, 0] := 1; T[n_, k_] := Sum[Binomial[n, j] * Binomial[k*n + j - 1, n - 1], {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 24 2020 *)

T(n,k) = (1/k) * [x^n] ( (1 + x)/(1 - x) )^(k*n) for k > 0 and n > 0.
T(n,k) = Sum_{j=0..n} binomial(n,j) * binomial(k*n+j-1,n-1).
T(n,k) = (1/k) * Sum_{j=0..n} binomial(k*n,n-j) * binomial(k*n+j-1,j) for k > 0 and n > 0.
T(n,k) = Sum_{j=1..n} 2^j * binomial(n,j) * binomial(k*n-1,j-1) for n > 0.
T(n,k) = binomial(k*n-1, n-1)*hypergeom([-n, k*n], [1+(k-1)*n], -1) for k > 0. - Stefano Spezia, Aug 09 2025

cp's OEIS Frontend

A336537 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n^2+k+1,n)/(n^2+k+1).

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A336521 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) is the coefficient of x^(k*n) in expansion of ( (1 + x)/(1 - x) )^n.

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