A245692 Number T(n,k) of endofunctions f on [n] that are self-inverse on [k] but not on [k+1]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
1, 0, 1, 1, 1, 2, 12, 7, 4, 4, 144, 62, 28, 12, 10, 2000, 695, 264, 100, 40, 26, 32400, 9504, 3126, 1050, 370, 130, 76, 605052, 154007, 44716, 13458, 4256, 1366, 456, 232, 12845056, 2891776, 751872, 204776, 58784, 17292, 5272, 1624, 764
Offset: 0
A185391 a(n) = Sum_{k=0..n} A185390(n,k) * k.
0, 1, 10, 114, 1556, 25080, 468462, 9971920, 238551336, 6339784320, 185391061010, 5917263922944, 204735466350780, 7633925334590464, 305188474579874550, 13023103577435351040, 590850477768105474128, 28401410966866912051200, 1441935117039649859464986
Offset: 0
Keywords
Comments
The total number of elements, x in the domain of definition of all partial functions on n labeled objects such that for all i in {1,2,3,...} (f^i)(x) is defined.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..385
Programs
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Mathematica
nn=20; tx=Sum[n^(n-1) x^n/n!,{n,1,nn}]; txy=Sum[n^(n-1) (x y)^n/n!, {n,1,nn}]; f[list_] := Select[list, #>0&]; D[Range[0,nn]! CoefficientList[Series[Exp[tx]/(1-txy),{x,0,nn}],x],y]/.y->1 -
PARI
{a(n) = (n+1)^(n+1)-sum(k=1, n+1, binomial(n+1, k)*k^k*(n+1-k)^(n+1-k))/(n+1)} \\ Seiichi Manyama, Jun 01 2019
Formula
a(n) = (n+1)^(n+1) - A001865(n+1). - Seiichi Manyama, Jun 01 2019
A262458 Number of possible conjunction tables in the n-valued logics.
1, 1, 256, 14348907, 281474976710656, 2910383045673370361328125, 22452257707354557240087211123792674816, 174251498233690814305510551794710260107945042018748343, 1766847064778384329583297500742918515827483896875618958121606201292619776
Offset: 1
Comments
In a conjunction table, the only requirements are (i) True ^ True = True; and (ii) no other conjunction gives True. So each of the other n^2-1 entries can be assigned any of n-1 non-True values, giving raise to the formula (n-1)^(n^2-1).
References
- H. Eves, C. V. Newsom, An Introduction to the Foundation and Fundamental concepts of Mathematics, Rinehart & Co., New York, 1958.
Programs
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PARI
a(n)=(n-1)^(n^2-1) \\ Charles R Greathouse IV, Sep 23 2015
Formula
a(n) = (n-1)^(n^2-1).
A262459 Number of n-valued logics.
0, 1, 3072, 2066242608, 562949953421312000, 94296410679817199707031250000, 13584783430350289567229247264873005482770432, 2238270252895659596439884146393953794861120109653175515742208
Offset: 1
References
- H. Eves and C. V. Newsom, An Introduction to the Foundation and Fundamental concepts of Mathematics, Rinehart & Co., New York, 1958.
Links
- MathWorld, Three-Valued Logic
Programs
-
PARI
a(n) = (n-1)^(n^2+1) * n^(n-2) \\ Charles R Greathouse IV, Sep 23 2015
Comments
Examples
T(3,1) = 7: (1,1,1), (1,1,2), (1,1,3), (1,3,1), (1,3,3), (3,1,1), (3,3,1). T(3,2) = 4: (1,2,1), (1,2,2), (2,1,1), (2,1,2). T(3,3) = 4: (1,2,3), (1,3,2), (2,1,3), (3,2,1). Triangle T(n,k) begins: 0 : 1; 1 : 0, 1; 2 : 1, 1, 2; 3 : 12, 7, 4, 4; 4 : 144, 62, 28, 12, 10; 5 : 2000, 695, 264, 100, 40, 26; 6 : 32400, 9504, 3126, 1050, 370, 130, 76; 7 : 605052, 154007, 44716, 13458, 4256, 1366, 456, 232; ...Links
Crossrefs
Programs
Maple
g:= proc(n) g(n):= `if`(n<2, 1, g(n-1)+(n-1)*g(n-2)) end: H:= (n, k)-> add(binomial(n-k, i)*binomial(k, i)*i!* g(k-i)*n^(n-k-i), i=0..min(k, n-k)): T:= (n, k)-> H(n, k) -H(n, k+1): seq(seq(T(n, k), k=0..n), n=0..10);Mathematica
g[n_] := g[n] = If[n<2, 1, g[n-1] + (n-1)*g[n-2]]; H[0, 0] = 1; H[n_, k_] := Sum[Binomial[n-k, i]*Binomial[k, i]*i!*g[k-i]*n^(n-k-i), {i, 0, Min[k, n-k]}]; T[n_, k_] := H[n, k] - H[n, k+1]; Table[T[n, k], {n, 0, 10}, { k, 0, n}] // Flatten (* Jean-François Alcover, Feb 19 2017, translated from Maple *)Formula