A361533 -id:A361533 - 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

A293049 Expansion of e.g.f. exp(x^3/(1 - x)).

Original entry on oeis.org

1, 0, 0, 6, 24, 120, 1080, 10080, 100800, 1149120, 14515200, 199584000, 2973801600, 47740492800, 820928908800, 15049152518400, 292919058432000, 6031865968128000, 130990787582054400, 2991455760887193600, 71659101232502784000, 1796424431562528768000

Offset: 0

Seiichi Manyama, Sep 29 2017

nonn

For n > 4, a(n) is a multiple of 10. - Muniru A Asiru, Oct 09 2017

Seiichi Manyama, Table of n, a(n) for n = 0..444

Column k=2 of A293053.
E.g.f.: Product_{i>k} exp(x^i): A000262 (k=0), A052845 (k=1), this sequence (k=2), A293050 (k=3).
Cf. A361533, A361572.

a:= proc(n) option remember; `if`(n=0, 1, add(
      a(n-j)*binomial(n-1, j-1)*j!, j=3..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 30 2017
seq(factorial(k)*coeftayl(exp(x^3/(1-x)), x = 0, k),k=0..50); # Muniru A Asiru, Oct 09 2017

CoefficientList[Series[E^(x^3/(1-x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Sep 30 2017 *)

x='x+O('x^66); Vec(serlaplace(exp(x^3/(1-x))))

E.g.f.: Product_{i>2} exp(x^i).
a(n) ~ n^(n-1/4) * exp(-5/2 + 2*sqrt(n) - n) / sqrt(2). - Vaclav Kotesovec, Sep 30 2017
a(n) = 2*(n-1) * a(n-1) - (n-1)*(n-2) * a(n-2) + 6*binomial(n-1,2) * a(n-3) - 12*binomial(n-1,3) * a(n-4) for n > 3. - Seiichi Manyama, Mar 15 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)/k!.
a(0) = 1; a(n) = (n-1)! * 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

A361547 Expansion of e.g.f. exp(x^5/(120 * (1-x))).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 6, 42, 336, 3024, 30366, 335412, 4041576, 52756704, 741620880, 11169844686, 179448036768, 3063069801792, 55360031126400, 1056123043335360, 21208345049147256, 447183762148547424, 9877939209960101280, 228112734232663600320

Offset: 0

Seiichi Manyama, Mar 15 2023

In general, if m>=1 and e.g.f. = exp(x^m / (m! * (1-x))), then a(n) ~ n! * exp(2*sqrt(n/m!) - (2*m-1)/(2*m!)) / (2*sqrt(Pi) * m!^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 28 2025

Cf. A185369, A361533, A361545.

RecurrenceTable[{4 (-5 + n) (-4 + n) (-3 + n) (-2 + n) (-1 + n) a[-6 + n] - 5 (-4 + n) (-3 + n) (-2 + n) (-1 + n) a[-5 + n] + 120 (-2 + n) (-1 + n) a[-2 + n] - 240 (-1 + n) a[-1 + n] + 120 a[n] == 0, a[1] == 0, a[2] == 0, a[3] == 0, a[4] == 0, a[5] == 1, a[6] == 6}, a, {n, 0, 25}] (* Vaclav Kotesovec, Aug 28 2025 *)

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x^5/(120*(1-x)))))

a(n) = 2*(n-1) * a(n-1) - (n-1)*(n-2) * a(n-2) + binomial(n-1,4) * a(n-5) - 4*binomial(n-1,5) * a(n-6) for n > 5.
From Seiichi Manyama, Jun 17 2024: (Start)
a(n) = n! * Sum_{k=0..floor(n/5)} binomial(n-4*k-1,n-5*k)/(120^k * k!).
a(0) = 1; a(n) = ((n-1)!/120) * Sum_{k=5..n} k * a(n-k)/(n-k)!. (End)
a(n) ~ 2^(-5/4) * 15^(-1/4) * exp(-3/80 + sqrt(n/30) - n) * n^(n - 1/4). - Vaclav Kotesovec, Aug 28 2025

A373757 Expansion of e.g.f. exp(x^3/(6 * (1 - x)^2)).

Original entry on oeis.org

1, 0, 0, 1, 8, 60, 490, 4480, 45920, 524440, 6619200, 91568400, 1377884200, 22401579200, 391192401600, 7300174281400, 144938169376000, 3049711320656000, 67777255079934400, 1586172656920051200, 38984454900431040000, 1003827897443395024000

Offset: 0

Seiichi Manyama, Jun 17 2024

nonn

Cf. A361533, A361573.

With[{nn=30},CoefficientList[Series[Exp[x^3/(6(1-x)^2)],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Dec 30 2024 *)

a(n) = n!*sum(k=0, n\3, binomial(n-k-1, n-3*k)/(6^k*k!));

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, j*(j-2)*v[i-j+1]/(i-j)!)); v;

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-k-1,n-3*k)/(6^k * k!).

a(0) = 1; a(n) = ((n-1)!/6) * Sum_{k=3..n} k * (k-2) * a(n-k)/(n-k)!.

A373772 Expansion of e.g.f. exp(x^3 / (6 * (1 - x))) / (1 - x).

Original entry on oeis.org

1, 1, 2, 7, 32, 180, 1210, 9450, 84000, 836920, 9234400, 111742400, 1471023400, 20925905000, 319830310800, 5226116295400, 90906373958400, 1676967192700800, 32697692264036800, 671856896755844800, 14509136903381120000, 328520930667097168000

Offset: 0

Seiichi Manyama, Jun 18 2024

nonn

Cf. A130906, A361597, A373773.

Cf. A000930, A361533.

a(n) = n!*sum(k=0, n\3, binomial(n-2*k, n-3*k)/(6^k*k!));

a(n) = n! * Sum_{k=0..floor(n/3)} binomial(n-2*k,n-3*k)/(6^k * k!).

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

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A293049 Expansion of e.g.f. exp(x^3/(1 - x)).

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

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A361547 Expansion of e.g.f. exp(x^5/(120 * (1-x))).

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A373757 Expansion of e.g.f. exp(x^3/(6 * (1 - x)^2)).

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A373772 Expansion of e.g.f. exp(x^3 / (6 * (1 - x))) / (1 - x).

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