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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Paulius Dilkas

Paulius Dilkas's wiki page.

Paulius Dilkas has authored 2 sequences.

A366672 a(n) = A002720(n)^n.

Original entry on oeis.org

1, 2, 49, 39304, 1908029761, 8831763846882976, 5602661527604399327549089, 659308109505417338723017914068713088, 18666765602783048904120522995911258148623099215361, 159740893387079678500933964995221201596055121928224632284394525184
Offset: 0

Views

Author

Paulius Dilkas, Oct 15 2023

Keywords

Comments

This is the model count of the following sentence in first-order logic:
(forall w, x, y, z. P(x, y, z) /\ P(w, y, z) => x = w) /\
(forall w, x, y, z. P(x, y, z) /\ P(x, w, z) => y = w).

Examples

			When n = 2, i.e., the domain is [2] = {1, 2}, both P(x, y, 1) and P(x, y, 2) represent partial injective functions from [2] to [2]. Since there are seven such functions, a(n) = 7^2 = 49.

Links

Crossrefs

Cf. A002720, A036740.

Programs

  • Mathematica
    Table[(n!*LaguerreL[n, -1])^n, {n, 0, 10}] (* Vaclav Kotesovec, Oct 20 2023 *)

Formula

a(n) ~ n^(n*(n + 1/4)) / (2^(n/2) * exp(n^2 - 2*n^(3/2) + n/2 - 31*sqrt(n)/48 + 17/192)) * (1 - 281/(5120*sqrt(n)) + 3074161/(52428800*n)). - Vaclav Kotesovec, Oct 20 2023

A360820 a(n) = Sum_{k=0..n} binomial(n, k)*2^(n^2-k*(n-k)).

Original entry on oeis.org

1, 4, 48, 1792, 221184, 98566144, 173946175488, 1281755680079872, 39534286378918477824, 5018464395368794081460224, 2586745980900067184722499862528, 5375203895735606878055792019528220672, 44865714160227204455469409035569750630989824, 1501355804811017489524770237231795462175548447391744
Offset: 0

Views

Author

Paulius Dilkas, Feb 21 2023

Keywords

Comments

This is the model count of the following sentence in first-order logic: forall X. forall Y. friends(X, Y) /\ smokes(X) -> smokes(Y).

Examples

			If the domain is {1}, then all four interpretations ({}, {smokes(1)}, {friends(1, 1)}, {smokes(1), friends(1, 1)}) are models, so a(1) = 4.

Links

Programs

  • Mathematica
    a[n_] := Sum[Binomial[n, k]*2^(n^2 - k*(n - k)), {k, 0, n}]; Array[a, 14, 0] (* Amiram Eldar, Feb 24 2023 *)
  • PARI
    a(n) = sum(k=0, n, binomial(n, k)*2^(n^2-k*(n-k))); \\ Michel Marcus, Feb 22 2023

Formula

a(n) = Sum_{k=0..n} binomial(n, k)*2^(n^2-k*(n-k)).
a(n) ~ 2^(n^2 + 1). - Vaclav Kotesovec, Feb 22 2023

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