A307883 -id:A307883 - cp's OEIS frontend

A335309 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * n^(n-k).

1, 3, 22, 245, 3606, 65527, 1411404, 35066313, 985483270, 30869546411, 1065442493556, 40144438269949, 1638733865336764, 72012798200637855, 3388250516614331416, 169894851136173584145, 9041936334960057699654, 508945841697238471315027, 30202327515992972576218980

Offset: 0

Ilya Gutkovskiy, May 31 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..381

Cf. A001850, A069835, A084771, A084772, A098659, A187021, A307883, A331656, A335310.

Join[{1}, Table[Sum[Binomial[n, k] Binomial[n + k, k] n^(n - k), {k, 0, n}], {n, 1, 18}]]
Table[SeriesCoefficient[1/Sqrt[1 - 2 (n + 2) x + n^2 x^2], {x, 0, n}], {n, 0, 18}]
Table[n! SeriesCoefficient[Exp[(n + 2) x] BesselI[0, 2 Sqrt[n + 1] x], {x, 0, n}], {n, 0, 18}]
Table[Hypergeometric2F1[-n, -n, 1, 1 + n], {n, 0, 18}]

a(n) = sum(k=0, n, binomial(n,k)^2*(n+1)^k); \\ Michel Marcus, Jun 01 2020

a(n) = central coefficient of (1 + (n + 2)*x + (n + 1)*x^2)^n.
a(n) = [x^n] 1 / sqrt(1 - 2*(n + 2)*x + n^2*x^2).
a(n) = n! * [x^n] exp((n + 2)*x) * BesselI(0,2*sqrt(n + 1)*x).
a(n) = Sum_{k=0..n} binomial(n,k)^2 * (n+1)^k.
a(n) ~ exp(2*sqrt(n)) * n^(n - 1/4) / (2*sqrt(Pi)) * (1 + 11/(12*sqrt(n))). - Vaclav Kotesovec, Jan 09 2023

A336187 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j * binomial(n,j)^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 13, 8, 1, 1, 5, 34, 63, 16, 1, 1, 6, 81, 352, 321, 32, 1, 1, 7, 186, 1685, 3946, 1683, 64, 1, 1, 8, 421, 7416, 38401, 46744, 8989, 128, 1, 1, 9, 946, 30835, 328146, 963525, 573616, 48639, 256, 1, 1, 10, 2113, 122816, 2590225, 16971876, 25346385, 7217536, 265729, 512, 1

Offset: 0

Seiichi Manyama, Jul 11 2020

Column k is the diagonal of the rational function 1 / (Product_{j=1..k} (1-x_j) - k * Product_{j=1..k} x_j) for k>0.

			Square array begins:
  1,  1,    1,     1,      1,        1, ...
  1,  2,    3,     4,      5,        6, ...
  1,  4,   13,    34,     81,      186, ...
  1,  8,   63,   352,   1685,     7416, ...
  1, 16,  321,  3946,  38401,   328146, ...
  1, 32, 1683, 46744, 963525, 16971876, ...

Columns k=0-3 give: A000012, A000079, A001850, A206180.
Main diagonal gives A336188.
Cf. A307883, A336163.

Unprotect[Power]; 0^0 = 1; T[n_, k_] := Sum[ k^j*Binomial[n, j]^k, {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 11 2020 *)

A336163 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j * binomial(n,j)^3.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 10, 1, 1, 4, 21, 56, 1, 1, 5, 34, 171, 346, 1, 1, 6, 49, 352, 1521, 2252, 1, 1, 7, 66, 605, 3946, 14283, 15184, 1, 1, 8, 85, 936, 8065, 46744, 138909, 104960, 1, 1, 9, 106, 1351, 14346, 113525, 573616, 1385163, 739162, 1, 1, 10, 129, 1856, 23281, 231876, 1656145, 7217536, 14072193, 5280932, 1

Offset: 0

Seiichi Manyama, Jul 10 2020

Column k is the diagonal of the rational function 1 / (1 + y + z + x*y + y*z + k*z*x + (k+1)*x*y*z).

Column k is the diagonal of the rational function 1 / ((1-x)*(1-y)*(1-z) - k*x*y*z).

			Square array begins:
  1,    1,     1,     1,      1,      1, ...
  1,    2,     3,     4,      5,      6, ...
  1,   10,    21,    34,     49,     66, ...
  1,   56,   171,   352,    605,    936, ...
  1,  346,  1521,  3946,   8065,  14346, ...
  1, 2252, 14283, 46744, 113525, 231876, ...

Columns k=0-6 give: A000012, A000172, A206178, A206180, A216483, A216636, A216698.
Main diagonal gives A241247.
Cf. A307883, A336179, A336187.

Unprotect[Power]; 0^0 = 1; T[n_, k_] := Sum[k^j * Binomial[n, j]^3, {j, 0, n}]; Table[T[k, n - k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Jul 11 2020 *)

cp's OEIS Frontend

A335309 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(n+k,k) * n^(n-k).

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A336187 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j * binomial(n,j)^k.

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A336163 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Sum_{j=0..n} k^j * binomial(n,j)^3.

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