A335344 -id:A335344 - 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

A361567 Expansion of e.g.f. exp(x^2/2 * (1+x)^2).

Original entry on oeis.org

1, 0, 1, 6, 15, 60, 555, 3150, 17745, 158760, 1399545, 10914750, 102920895, 1104323220, 11249313075, 119330961750, 1426411411425, 17429852840400, 213417453474225, 2791671804271350, 38524272522310575, 537569719902715500, 7732658753799054075

Offset: 0

Seiichi Manyama, Mar 16 2023

nonn

Cf. A047974, A361568, A361569.

Cf. A335344, A361278.

Table[n! * Sum[Binomial[2*k,n-2*k]/(2^k * k!), {k,0,n/2}], {n,0,20}] (* Vaclav Kotesovec, Mar 25 2023 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^2/2*(1+x)^2)))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/2*sum(j=2, i, j*binomial(2, j-2)*v[i-j+1]/(i-j)!)); v;

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(2*k,n-2*k)/(2^k * k!).
a(0) = 1; a(n) = ((n-1)!/2) * Sum_{k=2..n} k * binomial(2,k-2) * a(n-k)/(n-k)!.
From Vaclav Kotesovec, Mar 25 2023: (Start)
a(n) = (n-1)*a(n-2) + 3*(n-2)*(n-1)*a(n-3) + 2*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ 2^(n/4 - 1) * exp(1/128 - 3*2^(-29/4)*n^(1/4) - sqrt(n/2)/16 + 2^(-3/4)*n^(3/4) - 3*n/4) * n^(3*n/4). (End)

A361596 Expansion of e.g.f. exp( x^2/(2 * (1-x)^2) ) / (1-x).

Original entry on oeis.org

1, 1, 3, 15, 99, 795, 7485, 80745, 981225, 13253625, 196834995, 3185662095, 55770765435, 1049572599075, 21120725230605, 452384160453225, 10272547048388625, 246434674107647025, 6226347228582355875, 165224032352989584975, 4593512876411509125075

Offset: 0

Seiichi Manyama, Mar 16 2023

nonn

Cf. A002720, A361597.

Cf. A335344.

Table[n! * Sum[Binomial[n,2*k]/(2^k * k!), {k,0,n/2}], {n,0,20}] (* Vaclav Kotesovec, Mar 17 2023 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^2/(2*(1-x)^2))/(1-x)))

a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n,2*k)/(2^k * k!).
From Vaclav Kotesovec, Mar 17 2023: (Start)
a(n) = (3*n - 2)*a(n-1) - (n-1)*(3*n - 5)*a(n-2) + (n-2)^2*(n-1)*a(n-3).
a(n) ~ 3^(-1/2) * exp(1/6 - n^(1/3)/2 + 3*n^(2/3)/2 - n) * n^(n + 1/6) * (1 + 49/(108*n^(1/3)) + 3293/(116640*n^(2/3))). (End)

A335345 Expansion of e.g.f. exp(x^2/(2*(1 - x)^3)).

Original entry on oeis.org

1, 0, 1, 9, 75, 690, 7305, 89145, 1237425, 19221300, 329371245, 6157738125, 124551652995, 2707913238030, 62945320162725, 1557291398788125, 40844991621859425, 1131753403094113800, 33025920511859300025, 1012128709342410284625, 32494107983067177522075

Offset: 0

Ilya Gutkovskiy, Jun 02 2020

nonn

Cf. A001809, A091695, A185369, A335344.

nmax = 20; CoefficientList[Series[Exp[x^2/(2 (1 - x)^3)], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = (1/4) Sum[Binomial[n - 1, k - 1] k (k - 1) k! a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]

seq(n)=Vec(serlaplace(exp(x^2/(2*(1 - x)^3) + O(x*x^n)))) \\ Andrew Howroyd, Jun 02 2020

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * A001809(k) * a(n-k).
D-finite with recurrence 2*a(n) +8*(-n+1)*a(n-1) +2*(n-1)*(6*n-13)*a(n-2) -(n-1)*(n-2)*(8*n-23)*a(n-3) +2*(n-1)*(n-2)*(n-3)*(n-4)*a(n-4)=0. - R. J. Mathar, Jun 05 2020
a(n) ~ 2^(-9/8) * 3^(1/8) * n^(n - 1/8) * exp(1/54 - n^(1/4)/(2^(15/4)*3^(5/4)) - sqrt(6*n)/12 + 2^(7/4)*3^(-3/4)*n^(3/4) - n). - Vaclav Kotesovec, Jun 11 2020
a(n) = n! * Sum_{k=0..floor(n/2)} binomial(n+k-1,n-2*k)/(2^k * k!). - Seiichi Manyama, Jun 17 2024

A361573 Expansion of e.g.f. exp(x^3/(6 * (1 - x)^3)).

Original entry on oeis.org

1, 0, 0, 1, 12, 120, 1210, 13020, 152880, 1975960, 28148400, 440470800, 7525441000, 139375236000, 2778421245600, 59239029249400, 1343609515248000, 32274288638592000, 818014942318974400, 21809788600885084800, 610079100418595808000, 17863467401461938256000

Offset: 0

Seiichi Manyama, Mar 16 2023

nonn

Cf. A000262, A335344, A361577.

With[{nn=30},CoefficientList[Series[Exp[x^3/(6(1-x)^3)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, May 20 2023 *)

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

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/6*sum(j=3, i, (-1)^(j-3)*j*binomial(-3, j-3)*v[i-j+1]/(i-j)!)); v;

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-1,n-3*k)/(6^k * k!).
a(0) = 1; a(n) = ((n-1)!/6) * Sum_{k=3..n} (-1)^(k-3) * k * binomial(-3,k-3) * a(n-k)/(n-k)!.
a(n) ~ 2^(-9/8) * exp(-1/24 + 5*2^(1/4)*n^(1/4)/48 - sqrt(2*n)/4 + 2^(7/4)*n^(3/4)/3 - n) * n^(n - 1/8) * (1 - 1203*2^(3/4)/(10240*n^(1/4))). - Vaclav Kotesovec, Mar 29 2023

A361577 Expansion of e.g.f. exp(x^4/(24 * (1 - x)^4)).

Original entry on oeis.org

1, 0, 0, 0, 1, 20, 300, 4200, 58835, 849240, 12814200, 203742000, 3430355775, 61363001700, 1168815948300, 23734579869000, 513878948207625, 11850279026586000, 290440507342986000, 7543064638441332000, 206860683821114948625, 5968372055889205462500

Offset: 0

Seiichi Manyama, Mar 16 2023

nonn

Cf. A000262, A335344, A361573.

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

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!/24*sum(j=4, i, (-1)^(j-4)*j*binomial(-4, j-4)*v[i-j+1]/(i-j)!)); v;

a(n) = n! * Sum_{k=0..floor(n/4)} binomial(n-1,n-4*k)/(24^k * k!).

a(0) = 1; a(n) = ((n-1)!/24) * Sum_{k=4..n} (-1)^(k-4) * k * binomial(-4,k-4) * a(n-k)/(n-k)!.

A366155 Expansion of e.g.f. exp(x^3/(3*(1-x)^3)).

Original entry on oeis.org

1, 0, 0, 2, 24, 240, 2440, 26880, 329280, 4518080, 69148800, 1168675200, 21564188800, 430048819200, 9195964377600, 209593877292800, 5068718054400000, 129599032442880000, 3492894468128665600, 98968805893769011200, 2940975338620999680000, 91452266705317726208000, 2969664371124258103296000

Offset: 0

Enrique Navarrete, Oct 01 2023

For n>0, a(n) is the number of ways to split n people into nonempty groups, have each group sit around a circular table, and select 3 people from each table (where two seating arrangements are considered identical if each person has the same left neighbors in both of them).
2*A001754(n) is the number of ways to seat n persons around a circular table and select 3 of them if only one table is used.
A335344 is the corresponding sequence if 2 persons are selected from each table, and A000262 if only one person is selected from each table.

			a(7)=26880 since, using one table, there are 6! circular seatings and binomial(7,3) ways to select 3 persons, hence 25200 ways. Using two tables, the only way we can select 3 persons from each one is seating 4 persons in one table and 3 in the other, which can be done in 420 ways; then choosing 3 persons from each table can be done in 4 ways, for a total of 1680 ways; hence 25200 + 1680 = 26880.

Cf. A000262, A001754, A335344.

CoefficientList[Series[Exp[x^3/(3*(1-x)^3)],{x,0,22}],x]Table[n!,{n,0,22}] (* Stefano Spezia, Oct 02 2023 *)
nmax = 25; Join[{1}, Table[n!*Sum[Binomial[n-1, 3*k-1]/(3^k*k!), {k, 1, n}], {n, 1, nmax}]] (* Vaclav Kotesovec, Aug 28 2025 *)

From Vaclav Kotesovec, Aug 28 2025: (Start)
For n > 0, a(n) = n! * Sum_{k=1..n} binomial(n-1, 3*k-1) / (3^k * k!).
a(n) = 4*(n-1)*a(n-1) - 6*(n-2)*(n-1)*a(n-2) + (n-2)*(n-1)*(4*n-11)*a(n-3) - (n-4)*(n-3)*(n-2)*(n-1)*a(n-4).
a(n) ~ exp(-1/12 + 5*n^(1/4)/24 - n^(1/2)/2 + 4*n^(3/4)/3 - n) * n^(n - 1/8) / 2 * (1 - 563/(2560*n^(1/4))). (End)

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

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A361567 Expansion of e.g.f. exp(x^2/2 * (1+x)^2).

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A361596 Expansion of e.g.f. exp( x^2/(2 * (1-x)^2) ) / (1-x).

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A335345 Expansion of e.g.f. exp(x^2/(2*(1 - x)^3)).

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A361573 Expansion of e.g.f. exp(x^3/(6 * (1 - x)^3)).

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A361577 Expansion of e.g.f. exp(x^4/(24 * (1 - x)^4)).

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A366155 Expansion of e.g.f. exp(x^3/(3*(1-x)^3)).

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