A361547 -id:A361547 - cp's OEIS frontend

A185369 Number of simple labeled graphs on n nodes of degree 1 or 2 without cycles.

1, 0, 1, 3, 15, 90, 645, 5355, 50505, 532980, 6219045, 79469775, 1103335695, 16533226710, 265888247625, 4566885297975, 83422361847825, 1614626682669000, 33003508539026025, 710350201433547675, 16057073233633006575

Offset: 0

Geoffrey Critzer, Feb 20 2011

nonn

			a(4) = 15 because there are 15 simple labeled graphs on 4 nodes of degree 1 or 2 without cycles: 1-2 3-4, 1-3 2-4, 1-4 2-3, 1-2-3-4, 1-2-4-3, 1-3-2-4, 1-3-4-2, 1-4-2-3, 1-4-3-2, 2-1-3-4, 2-1-4-3, 3-1-2-4, 3-1-4-2, 4-1-2-3, 4-1-3-2.

Herbert S. Wilf, Generatingfunctionology, p. 104.

Alois P. Heinz, Table of n, a(n) for n = 0..400

Cf. A335344, A335345.

Cf. A361533, A361545, A361547.

a:= proc(n) option remember;
       `if`(n<2, 1-n, add(binomial(n-1, k-1) *k!/2 *a(n-k), k=2..n))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 24 2011

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

E.g.f.: exp(1/(2*(1-x))-x/2-1/2).
a(n) = 1-n if n<2, else a(n) = Sum_{k=2..n} C(n-1,k-1) * k!/2 * a(n-k).
a(n) ~ 2^(-3/4)*n^(n-1/4)*exp(-3/4+sqrt(2*n)-n). - Vaclav Kotesovec, Sep 25 2013
Conjecture: +2*a(n) +4*(-n+1)*a(n-1) +2*(n-1)*(n-3)*a(n-2) +(n-1)*(n-2)*a(n-3)=0. - R. J. Mathar, Jun 14 2016
a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n-k-1,n-2*k)/(2^k * k!). - Seiichi Manyama, Jun 17 2024

A361533 Expansion of e.g.f. exp(x^3/(6 * (1-x))).

Original entry on oeis.org

1, 0, 0, 1, 4, 20, 130, 980, 8400, 80920, 865200, 10164000, 130114600, 1802600800, 26867640800, 428661633400, 7288513232000, 131558835408000, 2512282795422400, 50600743739145600, 1071998968264224000, 23829055696093648000, 554524256514356128000

Offset: 0

Seiichi Manyama, Mar 15 2023

nonn

Cf. A185369, A361545, A361547.

Cf. A293049.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^3/(6*(1-x)))))

a(n) = 2*(n-1) * a(n-1) - (n-1)*(n-2) * a(n-2) + binomial(n-1,2) * a(n-3) - 2*binomial(n-1,3) * a(n-4) for n > 3.
a(n) ~ 2^(-3/4) * 3^(-1/4) * exp(-5/12 + sqrt(2*n/3) - n) * n^(n - 1/4). - Vaclav Kotesovec, Mar 29 2023
From Seiichi Manyama, Jun 17 2024: (Start)
a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k-1,n-3*k)/(6^k * k!).
a(0) = 1; a(n) = ((n-1)!/6) * Sum_{k=3..n} k * a(n-k)/(n-k)!. (End)

A361545 Expansion of e.g.f. exp(x^4/(24 * (1-x))).

Original entry on oeis.org

1, 0, 0, 0, 1, 5, 30, 210, 1715, 15750, 160650, 1801800, 22043175, 292116825, 4168464300, 63725161500, 1039028615625, 17998106626500, 330068683444500, 6388785205803000, 130156170633113625, 2783924007745505625, 62375052003905891250, 1460924768552182683750

Offset: 0

Seiichi Manyama, Mar 15 2023

Cf. A185369, A361533, A361547.

Cf. A293050.

RecurrenceTable[{3 (-4 + n) (-3 + n) (-2 + n) (-1 + n) a[-5 + n] - 4 (-3 + n) (-2 + n) (-1 + n) a[-4 + n] + 24 (-2 + n) (-1 + n) a[-2 + n] - 48 (-1 + n) a[-1 + n] + 24 a[n] == 0, a[1] == 0, a[2] == 0, a[3] == 0, a[4] == 1, a[5] == 5}, a, {n, 0, 25}] (* Vaclav Kotesovec, Aug 28 2025 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^4/(24*(1-x)))))

a(n) = 2*(n-1) * a(n-1) - (n-1)*(n-2) * a(n-2) + binomial(n-1,3) * a(n-4) - 3*binomial(n-1,4) * a(n-5) for n > 4.
From Seiichi Manyama, Jun 17 2024: (Start)
a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n-3*k-1,n-4*k)/(24^k * k!).
a(0) = 1; a(n) = ((n-1)!/24) * Sum_{k=4..n} k * a(n-k)/(n-k)!. (End)
a(n) ~ 2^(-5/4) * 3^(-1/4) * exp(-7/48 + sqrt(n/6) - n) * n^(n - 1/4). - Vaclav Kotesovec, Aug 28 2025

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A185369 Number of simple labeled graphs on n nodes of degree 1 or 2 without cycles.

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A361533 Expansion of e.g.f. exp(x^3/(6 * (1-x))).

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A361545 Expansion of e.g.f. exp(x^4/(24 * (1-x))).

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