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A336188
a(n) = Sum_{k=0..n} n^k * binomial(n,k)^n.
7
1, 2, 13, 352, 38401, 16971876, 29359436149, 207003074670848, 5679112509686022145, 636468045901197095750500, 277939985126193076692203962501, 494649880078824954885176565423811200, 3447375085398645453825889951638344722092289, 97424105704407389799712313421357308088296084669504
OFFSET
0,2
FORMULA
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
MATHEMATICA
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 *)
PROG
(PARI) {a(n) = sum(k=0, n, n^k*binomial(n, k)^n)}
(Magma) [(&+[n^j*Binomial(n, j)^n: j in [0..n]]): n in [0..20]]; // G. C. Greubel, Aug 26 2022
(SageMath) [sum(n^j*binomial(n, j)^n for j in (0..n)) for n in (0..20)] # G. C. Greubel, Aug 26 2022
CROSSREFS
Main diagonal of A336187.
Sequence in context: A304727 A355730 A075620 * A348015 A131525 A366722
KEYWORD
nonn,nice
AUTHOR
Seiichi Manyama, Jul 11 2020
STATUS
approved