A036774 -id:A036774 - cp's OEIS frontend

A201996 The number of endofunctions on n points such that all recurrent elements have at most 3 preimages and all nonrecurrent elements have at most 2 preimages.

1, 1, 4, 27, 252, 3000, 43380, 737730, 14419440, 318381840, 7835486400, 212634298800, 6307073942400, 202983948367200, 7044249755743200, 262198957638618000, 10419369722457696000, 440257835691561888000, 19709455059507717504000, 931885122471464345184000, 46401644730376725229440000

Offset: 0

Geoffrey Critzer, Dec 07 2011

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..390
Giulio Cerbai and Anders Claesson, Counting fixed-point-free Cayley permutations, arXiv:2507.09304 [math.CO], 2025. See p. 25.

a = (1 - x - (1 - 2 x - x^2)^(1/2))/x; Range[0, 20]! CoefficientList[Series[1/(1 - a), {x, 0, 20}], x]

E.g.f.: 1/(1-A(x)) where A(x) is the e.g.f. for A036774.

a(n) ~ n! * (5/2)^n/5. - Vaclav Kotesovec, Sep 24 2013

A200793 The number of forests on n nodes of rooted labeled binary trees (each node has degree <=2).

Original entry on oeis.org

1, 1, 3, 16, 121, 1191, 14461, 209098, 3510921, 67175461, 1443249271, 34412298636, 901898694313, 25775139581491, 797824620178041, 26592701386533766, 949705032131053201, 36181186751341438473, 1464760631695118359051, 62798619981256526628136

Offset: 0

Geoffrey Critzer, Nov 22 2011

nonn

u=(1-x-((x-1)^2-2x^2)^(1/2))/x; Range[0,20]! CoefficientList[Series[Exp[u],{x,0,20}],x]

E.g.f.: exp(A(x)) where A(x) is the e.g.f. for A036774.

a(n) ~ sqrt(2-sqrt(2)) * (1+sqrt(2))^(n+1) * exp(sqrt(2)-n) * n^(n-1). - Vaclav Kotesovec, Sep 25 2013

A336310 Sum of path lengths over all labeled rooted unordered binary trees.

Original entry on oeis.org

0, 0, 2, 24, 300, 4260, 69120, 1271340, 26233200, 601246800, 15171105600, 418203324000, 12509695598400, 403696590897600, 13982667790291200, 517482647165484000, 20381726051118432000, 851302665544050720000, 37587618060140244096000, 1749369290830388555328000, 85599487854917373617280000

Offset: 0

Geoffrey Critzer, Jul 17 2020

nonn

Cf. A336309, A036774 (row sums).

nn = 20; Range[0, nn]! CoefficientList[ Series[-(((-1 + Sqrt[1 - 2 z - z^2]) (-1 + z + Sqrt[1 - 2 z - z^2]))/(z (-1 + 2 z + z^2))), {z, 0, nn}], z]

my(z='z+O('z^25)); concat([0,0], Vec(serlaplace(((1 -sqrt(1 -2*z -z^2))*(1 -z -sqrt(1 -2*z -z^2)))/(z*(1 -2*z -z^2))))) \\ Joerg Arndt, Jul 18 2020

E.g.f.: ((1 -sqrt(1 -2*z -z^2))*(1 -z -sqrt(1 -2*z -z^2)))/(z*(1 -2*z -z^2)).
a(n) = Sum_{k} A336309(n,k)*k, for n>=1.
a(n) ~ n!/2 * (sqrt(2) + 1)^(n+1) * (1 - sqrt((10-sqrt(2))/(Pi*n))). - Vaclav Kotesovec, Jul 17 2020

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A201996 The number of endofunctions on n points such that all recurrent elements have at most 3 preimages and all nonrecurrent elements have at most 2 preimages.

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A200793 The number of forests on n nodes of rooted labeled binary trees (each node has degree <=2).

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A336310 Sum of path lengths over all labeled rooted unordered binary trees.

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