A336578 -id:A336578 - cp's OEIS frontend

A336575 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} 3^j * binomial(n,j) * binomial(k*n,j-1) for n > 0.

1, 1, 3, 1, 3, 3, 1, 3, 12, 3, 1, 3, 21, 57, 3, 1, 3, 30, 192, 300, 3, 1, 3, 39, 408, 2001, 1686, 3, 1, 3, 48, 705, 6402, 22539, 9912, 3, 1, 3, 57, 1083, 14799, 109137, 267276, 60213, 3, 1, 3, 66, 1542, 28488, 338430, 1964010, 3287496, 374988, 3, 1, 3, 75, 2082, 48765, 817743, 8181597, 36718680, 41556585, 2381322, 3

Offset: 0

Seiichi Manyama, Jul 26 2020

			Square array begins:
  1,    1,     1,      1,      1,      1, ...
  3,    3,     3,      3,      3,      3, ...
  3,   12,    21,     30,     39,     48, ...
  3,   57,   192,    408,    705,   1083, ...
  3,  300,  2001,   6402,  14799,  28488, ...
  3, 1686, 22539, 109137, 338430, 817743, ...

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

Columns k=0-4 give: A122553, A047891, A219535, A336538, A336540.
Main diagonal gives A336578.
Cf. A336534, A336573, A336574.

T[0, k_] := 1; T[n_, k_] := Sum[3^j * Binomial[n, j] * Binomial[k*n, j - 1], {j, 1, n}]/n; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 27 2020 *)

T(n, k) = if(n==0, 1, sum(j=1, n, 3^j*binomial(n, j)*binomial(k*n, j-1))/n);

T(n, k) = my(A=1+x*O(x^n)); for(i=0, n, A=1+x*A^k*(2+A)); polcoeff(A, n);

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

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

G.f. A_k(x) of column k satisfies A_k(x) = 1 + x * A_k(x)^k * (2 + A_k(x)).
T(n,k) = Sum_{j=0..n} 2^(n-j) * binomial(n,j) * binomial(k*n+j+1,n)/(k*n+j+1).
T(n,k) = (1/(k*n+1)) * Sum_{j=0..n} 2^j * binomial(k*n+1,j) * binomial((k+1)*n-j,n-j).
T(n,k) = (1/n) * Sum_{j=0..n-1} (-2)^j * 3^(n-j) * binomial(n,j) * binomial((k+1)*n-j,n-1-j) for n > 0. - Seiichi Manyama, Aug 10 2023
T(n,k) = 3*hypergeom([1-n, -k*n], [2], 3) for n > 0. - Stefano Spezia, Aug 09 2025

A336713 a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * binomial(n+(n-1)*k,k-1) for n > 0.

Original entry on oeis.org

1, 1, 1, 6, 76, 1447, 37206, 1212194, 47975271, 2238595055, 120453255172, 7347494056729, 501273291296174, 37833413358907566, 3130557361463956074, 281854137496597897755, 27433898122963009937892, 2870816347095046227070383, 321430790732030793454519088

Offset: 0

Seiichi Manyama, Aug 01 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..339

Main diagonal of A336708.

Cf. A336578, A335871, A336712, A336714.

a[0] = 1; a[n_] := Sum[(-1)^(n - k) * Binomial[n, k] * Binomial[n + (n - 1)*k, k - 1], {k, 1, n}] / n; Array[a, 19, 0] (* Amiram Eldar, Aug 01 2020 *)

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

a(n) ~ exp(n - 1/2 - exp(-1)) * n^(n - 5/2) / sqrt(2*Pi). - Vaclav Kotesovec, Aug 04 2025

A336714 a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} (-2)^(n-k) * binomial(n,k) * binomial(n+(n-1)*k,k-1) for n > 0.

Original entry on oeis.org

1, 1, 0, 2, 36, 766, 20910, 707472, 28740656, 1367040950, 74645106114, 4606416653654, 317237242964840, 24130334401571972, 2009783477119978508, 181958565624827141256, 17796032244661580019904, 1870078875109869688744870, 210155525478346375059816234, 25151873422906866362758095642

Offset: 0

Seiichi Manyama, Aug 01 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..339

Main diagonal of A336709.

Cf. A336578, A335871, A336712, A336713.

a[0] = 1; a[n_] := Sum[(-2)^(n - k) * Binomial[n, k] * Binomial[n + (n - 1)*k, k - 1], {k, 1, n}] / n; Array[a, 20, 0] (* Amiram Eldar, Aug 01 2020 *)

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

a(n) ~ exp(n - 1/2 - 2*exp(-1)) * n^(n - 5/2) / sqrt(2*Pi). - Vaclav Kotesovec, Aug 04 2025

A335871 a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} binomial(n,k) * binomial(n+(n-1)*k,k-1) for n > 0.

Original entry on oeis.org

1, 1, 3, 20, 234, 4159, 101538, 3182454, 122285201, 5575750271, 294529785168, 17697480642005, 1192398100081202, 89053864927236146, 7302988011333915878, 652439391227186881683, 63077327237347821501754, 6561701255914880362990927, 730833849063629052249986940

Offset: 0

Seiichi Manyama, Aug 01 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..339

Main diagonal of A336706.

Cf. A336578, A336712, A336713, A336714.

a[0] = 1; a[n_] := Sum[Binomial[n, k] * Binomial[n + (n - 1)*k, k - 1], {k, 1, n}] / n; Array[a, 19, 0] (* Amiram Eldar, Aug 01 2020 *)

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

A336712 a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} 2^(n-k) * binomial(n,k) * binomial(n+(n-1)*k,k-1) for n > 0.

Original entry on oeis.org

1, 1, 4, 30, 364, 6502, 158034, 4921112, 187897728, 8519286854, 447829041358, 26796275824186, 1798936842255128, 133933302810144684, 10953460639289615412, 976226180855018504472, 94181146038753255120480, 9778885058353578446996934, 1087326670244362420301889926

Offset: 0

Seiichi Manyama, Aug 01 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..339

Main diagonal of A336707.

Cf. A336578, A335871, A336713, A336714.

a[0] = 1; a[n_] := Sum[2^(n - k) * Binomial[n, k] * Binomial[n + (n - 1)*k, k - 1], {k, 1, n}] / n; Array[a, 19, 0] (* Amiram Eldar, Aug 01 2020 *)

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

A336577 a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * binomial(n^2+k+1,n)/(n^2+k+1).

Original entry on oeis.org

1, 3, 24, 498, 18708, 1055838, 80682414, 7829287392, 924359573112, 128815914107370, 20717986773639696, 3779867347688995698, 771666206195918154156, 174345811623642373266360, 43198501381068549879753648, 11648965476456962547182140512, 3396661425137920919866033312752

Offset: 0

Seiichi Manyama, Jul 26 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..297

Main diagonal of A336574.

Cf. A336495, A336537, A336578.

a[n_] := Sum[2^k * Binomial[n, k] * Binomial[n^2 + k + 1, n]/(n^2 + k + 1), {k, 0, n}];  Array[a, 17, 0] (* Amiram Eldar, Jul 27 2020 *)

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

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

a(n) = (1/(n^2+1)) * Sum_{k=0..n} 2^(n-k) * binomial(n^2+1,k) * binomial((n+1)*n-k,n-k).
a(n) ~ 3^n * exp(n + 1/6) * n^(n - 5/2) / sqrt(2*Pi). - Vaclav Kotesovec, Jul 31 2021
From Seiichi Manyama, Aug 10 2023: (Start)
a(n) = (1/n) * Sum_{k=0..n-1} (-1)^k * 3^(n-k) * binomial(n,k) * binomial((n+1)*n-k,n-1-k) for n > 0.
a(n) = (1/n) * Sum_{k=1..n} 3^k * 2^(n-k) * binomial(n,k) * binomial(n^2,k-1) for n > 0. (End)
a(n) = binomial(1+n^2, n)*hypergeom([-n, 1+n^2], [2-n+n^2], -2)/(1 + n^2). - Stefano Spezia, Aug 09 2025

cp's OEIS Frontend

A336575 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(0,k) = 1 and T(n,k) = (1/n) * Sum_{j=1..n} 3^j * binomial(n,j) * binomial(k*n,j-1) for n > 0.

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A336713 a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} (-1)^(n-k) * binomial(n,k) * binomial(n+(n-1)*k,k-1) for n > 0.

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A336714 a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} (-2)^(n-k) * binomial(n,k) * binomial(n+(n-1)*k,k-1) for n > 0.

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A335871 a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} binomial(n,k) * binomial(n+(n-1)*k,k-1) for n > 0.

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A336712 a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} 2^(n-k) * binomial(n,k) * binomial(n+(n-1)*k,k-1) for n > 0.

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A336577 a(n) = Sum_{k=0..n} 2^k * binomial(n,k) * binomial(n^2+k+1,n)/(n^2+k+1).

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