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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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS 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 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. Cf. A167010, A187021, A234971, A241247, A336202, A336213, A336214. Sequence in context: A304727 A355730 A075620 * A348015 A131525 A082751 Adjacent sequences: A336185 A336186 A336187 * A336189 A336190 A336191 KEYWORD nonn,nice AUTHOR Seiichi Manyama, Jul 11 2020 STATUS approved

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Last modified September 25 05:35 EDT 2023. Contains 365582 sequences. (Running on oeis4.)