A336187 -id:A336187 - cp's OEIS frontend

A336188 a(n) = Sum_{k=0..n} n^k * binomial(n,k)^n.

1, 2, 13, 352, 38401, 16971876, 29359436149, 207003074670848, 5679112509686022145, 636468045901197095750500, 277939985126193076692203962501, 494649880078824954885176565423811200, 3447375085398645453825889951638344722092289, 97424105704407389799712313421357308088296084669504

Offset: 0

Seiichi Manyama, Jul 11 2020

Seiichi Manyama, Table of n, a(n) for n = 0..57
Vaclav Kotesovec, Plot of a(n) / (2^((n+1)*(2*n-1)/2) * n^(log(n)/8) / Pi^((n-1)/2)) for n = 1..10000000

Main diagonal of A336187.

Cf. A167010, A187021, A234971, A241247, A336202, A336213, A336214.

[(&+[n^j*Binomial(n,j)^n: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 26 2022

Unprotect[Power]; 0^0 = 1; a[n_] := Sum[n^k * Binomial[n, k]^n, {k, 0, n} ]; Array[a, 14, 0] (* Amiram Eldar, Jul 11 2020 *)

{a(n) = sum(k=0, n, n^k*binomial(n, k)^n)}

[sum(n^j*binomial(n,j)^n for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 26 2022

Let f(n) = 2^((n+1)*(2*n-1)/2) * n^(log(n)/8) / Pi^((n-1)/2). For sufficiently large n 0.7675... < a(n)/f(n) < 0.7900... - Vaclav Kotesovec, Jul 11 2020
The above bounds of Vaclav Kotesovec can be recast as: |a(n)/f(n) - exp(-1/4)| <= (3*Pi)^(-2) for sufficiently large n. - Peter Luschny, Jul 12 2020
a(n) ~ exp(-1/4) * QPochhammer(exp(-4)) * QPochhammer(-n*exp(-2), exp(-4)) * 2^(n^2 + n/2) / Pi^(n/2) if n is even and a(n) ~ exp(-3/4) * QPochhammer(exp(-4)) * QPochhammer(-n*exp(-4), exp(-4)) * 2^(n^2 + n/2) * sqrt(n)  / Pi^(n/2) if n is odd. - Vaclav Kotesovec, Jul 13 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 *)

A336201 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, 0, 1, 1, -1, 0, 1, 1, -2, -3, 0, 1, 1, -3, -14, 11, 0, 1, 1, -4, -47, 136, 1, 0, 1, 1, -5, -134, 909, 106, -81, 0, 1, 1, -6, -347, 4736, 3585, -8492, 141, 0, 1, 1, -7, -846, 21655, 61906, -323523, 35344, 363, 0, 1, 1, -8, -1983, 91512, 771601, -8065624, 2201809, 395008, -1791, 0, 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, 0,  -1,    -2,      -3,       -4, ...
  1, 0,  -3,   -14,     -47,     -134, ...
  1, 0,  11,   136,     909,     4736, ...
  1, 0,   1,   106,    3585,    61906, ...
  1, 0, -81, -8492, -323523, -8065624, ...

Columns k=0-3 give: A000012, A000007, (-1)^n*A098332(n), A336182.
Main diagonal gives A336202.
Cf. A309010, A336179, A336187.

T[n_, k_] := Sum[If[k == j == 0, 1, (-k)^j] * Binomial[n, j]^k, {j, 0, n}]; Table[T[k, n-k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, May 01 2021 *)

cp's OEIS Frontend

A336188 a(n) = Sum_{k=0..n} n^k * binomial(n,k)^n.

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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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A336201 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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