A336202 -id:A336202 - 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

A336204 a(n) = Sum_{k=0..n} 2^k * binomial(n,k)^n.

Original entry on oeis.org

1, 3, 13, 171, 7761, 1256283, 741398869, 1609036666443, 13118066779885825, 399221556627301207443, 46476897754761801245056293, 20377119057713827002258336842283, 34592895120825704155462768381947657489, 222457046333769635263635086646525921070978443

Offset: 0

Seiichi Manyama, Jul 11 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..59

Main diagonal of A336203.

Cf. A167010, A336188, A336202, A336212.

[(&+[2^j*Binomial(n,j)^n: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 31 2022

Table[Sum[2^k*Binomial[n, k]^n, {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Jul 12 2020 *)

{a(n) = sum(k=0, n, 2^k*binomial(n, k)^n)};

def A336204(n): return sum(2^k*binomial(n,k)^n for k in (0..n))
[A336204(n) for n in (0..20)] # G. C. Greubel, Aug 31 2022

a(n) ~ c * 2^(n*(n+1)) / (Pi*n)^(n/2), where c = exp(-1/4) * Sum_{k = -oo..oo} 2^k * exp(-2*k^2) = 1.0434092897163574491113380912895917... if n is even and c = exp(-1/4) * Sum_{k = -oo..oo} 2^(k + 1/2) * exp(-2*(k + 1/2)^2) = 1.029587234777114329090639723058125257... if n is odd. - Vaclav Kotesovec, Jul 12 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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A336204 a(n) = Sum_{k=0..n} 2^k * binomial(n,k)^n.

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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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