A038205 -id:A038205 - cp's OEIS frontend

A098623 Consider the family of directed multigraphs enriched by the species of set partitions. Sequence gives number of those multigraphs with n labeled arcs.

1, 1, 8, 109, 2229, 62684, 2289151, 104344153, 5767234550, 378073098155, 28888082263581, 2536660090249102, 253007765488793325, 28383529110762969901, 3551558435250676339536, 492092920443604792460905, 75025155137863150912784409, 12516480979952118669729618300

Offset: 0

N. J. A. Sloane, Oct 26 2004

nonn

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

Andrew Howroyd, Table of n, a(n) for n = 0..200
G. Labelle, Counting enriched multigraphs according to the number of their edges (or arcs), Discrete Math., 217 (2000), 237-248.
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]

Cf. A000110, A098620, A098621, A098622.

\\ here R(n) is A000110 as e.g.f.
egfA020556(n)={my(bell=serlaplace(exp(exp(x + O(x^(2*n+1)))-1))); sum(i=0, n, sum(k=0, i, (-1)^k*binomial(i, k)*polcoef(bell, 2*i-k))*x^i/i!) + O(x*x^n)}
EnrichedGdSeq(R)={my(n=serprec(R, x)-1, B=subst(egfA020556(n), x, log(1+x + O(x*x^n)))); Vec(serlaplace(subst(B, x, R-polcoef(R,0))))}
R(n)={exp(exp(x + O(x*x^n))-1)}
EnrichedGdSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021

E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014505 and 1 + R(x) is the e.g.f. of A000110. - Andrew Howroyd, Jan 12 2021

Terms a(12) and beyond from Andrew Howroyd, Jan 12 2021

A014505 Number of digraphs with unlabeled (non-isolated) nodes and n labeled edges.

Original entry on oeis.org

1, 1, 6, 68, 1206, 29982, 981476, 40515568, 2044492988, 123175320988, 8697475219688, 709097832452880, 65934837808883016, 6920436929999656936, 812724019581549433520, 105986960037601701495680

Offset: 0

Simon Plouffe, gilbert(AT)lacim.uqam.ca (Gilbert Labelle)

nonn

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

G. Labelle, Counting enriched multigraphs according to the number of their edges (or arcs), Discrete Math., 217 (2000), 237-248.
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]

Cf. A014507.

E.g.f.: exp(-1) * Sum_{n>=0} (1+x)^(n^2-n) / n!. - Paul D. Hanna, Apr 25 2018

a(n) = n!*exp(-1) * Sum_{k>=sqrt(n)} binomial(k^2-k, n) / k!. - Paul D. Hanna, Apr 25 2018

A047865 Number of derangements of n where minimal cycle size is at least 4.

Original entry on oeis.org

1, 0, 0, 0, 6, 24, 120, 720, 6300, 58464, 586656, 6384960, 76471560, 994831200, 13939507296, 209097854784, 3345235180560, 56866395720960, 1023601917024000, 19448577603454464, 388972171805410656, 8168409582839579520, 179704944537482689920

Offset: 0

N. J. A. Sloane

nonn

H. S. Wilf, Generatingfunctionology, Academic Press, NY, 1990, p. 147, Eq. 5.2.9 (q=3).

Alois P. Heinz, Table of n, a(n) for n = 0..200
H. S. Wilf, Generatingfunctionology, 2nd edn., Academic Press, NY, 1994, p. 176, Eq. 5.2.9 (q=3).

Cf. A038205, A000166.

with(combstruct): ZL3:=[S,{S=Set(Cycle(Z,card>3))},labeled]:
seq (count (ZL3, size=n), n=0..21); # Zerinvary Lajos, Sep 26 2007

nn=20;Range[0,nn]!CoefficientList[Series[Exp[-x-x^2/2-x^3/3]/(1-x),{x,0,nn}],x]  (* Geoffrey Critzer, Nov 11 2012 *)

a(n) = (n-1)*a(n-1) + (n-1)*(n-2)*(n-3)*a(n-4).
E.g.f.: A(x) = 1/(1-x)*exp(-x-x^2/2-x^3/3) = 1 + 6*x^4/4! + 24*x^5/5! + ... satisfies the differential equation A'(x) = x^3/(1-x)*A(x). - Peter Bala, Apr 18 2012
a(n) ~ n! * exp(-11/6). - Vaclav Kotesovec, Aug 13 2013

Definition adjusted by Steven Finch, Mar 10 2022

A274760 The multinomial transform of A001818(n) = ((2*n-1)!!)^2.

Original entry on oeis.org

1, 1, 10, 478, 68248, 21809656, 13107532816, 13244650672240, 20818058883902848, 48069880140604832128, 156044927762422185270016, 687740710497308621254625536, 4000181720339888446834235653120, 29991260979682976913756629498334208

Offset: 0

Johannes W. Meijer, Jul 27 2016

nonn

The multinomial transform [MNL] transforms an input sequence b(n) into the output sequence a(n). Given the fact that the structure of the a(n) formulas, see the examples, lead to the multinomial coefficients A036039 the MNL transform seems to be an appropriate name for this transform. The multinomial transform is related to the exponential transform, see A274804 and the third formula. For the inverse multinomial transform [IML] see A274844.
To preserve the identity IML[MNL[b(n)]] = b(n) for n >= 0 for a sequence b(n) with offset 0 the shifted sequence b(n-1) with offset 1 has to be used as input for the MNL, otherwise information about b(0) will be lost in transformation.
In the a(n) formulas, see the examples, the multinomial coefficients A036039 appear.
We observe that a(0) = 1 and that this term provides no information about any value of b(n), this notwithstanding we will start the a(n) sequence with a(0) = 1.
The Maple programs can be used to generate the multinomial transform of a sequence. The first program uses the first formula which was found by Paul D. Hanna, see A158876, and Vladimir Kruchinin, see A215915. The second program uses properties of the e.g.f., see the sequences A158876, A213507, A244430 and A274539 and the third formula. The third program uses information about the inverse multinomial transform, see A274844.
Some MNL transform pairs are, n >= 1: A000045(n) and A244430(n-1); A000045(n+1) and A213527(n-1); A000108(n) and A213507(n-1); A000108(n-1) and A243953(n-1); A000142(n) and A158876(n-1); A000203(n) and A053529(n-1); A000110(n) and A274539(n-1); A000041(n) and A215915(n-1); A000035(n-1) and A177145(n-1); A179184(n) and A038205(n-1); A267936(n) and A000266(n-1); A267871(n) and A000090(n-1); A193356(n) and A088009(n-1).

			Some a(n) formulas, see A036039:
  a(0) = 1
  a(1) = 1*x(1)
  a(2) = 1*x(2) + 1*x(1)^2
  a(3) = 2*x(3) + 3*x(1)*x(2) + 1*x(1)^3
  a(4) = 6*x(4) + 8*x(1)*x(3) + 3*x(2)^2 + 6*x(1)^2*x(2) + 1*x(1)^4
  a(5) = 24*x(5) + 30*x(1)*x(4) + 20*x(2)*x(3) + 20*x(1)^2*x(3) + 15*x(1)*x(2)^2 + 10*x(1)^3*x(2) + 1*x(1)^5

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 1995, pp. 18-23.

M. Bernstein and N. J. A. Sloane, Some Canonical Sequences of Integers, arXiv:math/0205301 [math.CO], 2002; Linear Algebra and its Applications, Vol. 226-228 (1995), pp. 57-72. Erratum 320 (2000), 210. [Link to arXiv version]
M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
N. J. A. Sloane, Transforms.
Eric W. Weisstein MathWorld, Exponential Transform.

Cf. A274844, A274804, A036039, A001818, A001316, A215915, A008279.
Cf. A244430, A213527, A213507, A243953, A158876, A274539, A215915.
Cf. A090998, A008298, A271703, A112302, A135002, A274540.

nmax:= 13: b := proc(n): (doublefactorial(2*n-1))^2 end: a:= proc(n) option remember: if n=0 then 1 else add(((n-1)!/(n-k)!) * b(k) * a(n-k), k=1..n) fi: end: seq(a(n), n = 0..nmax); # End first MNL program.
nmax:=13: b := proc(n): (doublefactorial(2*n-1))^2 end: t1 := exp(add(b(n)*x^n/n, n = 1..nmax+1)): t2 := series(t1, x, nmax+1): a := proc(n): n!*coeff(t2, x, n) end: seq(a(n), n = 0..nmax); # End second MNL program.
nmax:=13: b := proc(n): (doublefactorial(2*n-1))^2 end: f := series(log(1+add(s(n)*x^n/n!, n=1..nmax)), x, nmax+1): d := proc(n): n*coeff(f, x, n) end: a(0) := 1: a(1) := b(1): s(1) := b(1): for n from 2 to nmax do s(n) := solve(d(n)-b(n), s(n)): a(n):=s(n): od: seq(a(n), n=0..nmax); # End third MNL program.

b[n_] := (2*n - 1)!!^2;
a[0] = 1; a[n_] := a[n] = Sum[((n-1)!/(n-k)!)*b[k]*a[n-k], {k, 1, n}];
Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Nov 17 2017 *)

a(n) = Sum_{k=1..n} ((n-1)!/(n-k)!)*b(k)*a(n-k), n >= 1 and a(0) = 1, with b(n) = A001818(n) = ((2*n-1)!!)^2.
a(n) = n!*P(n), with P(n) = (1/n)*(Sum_{k=0..n-1} b(n-k)*P(k)), n >= 1 and P(0) = 1, with b(n) = A001818(n) = ((2*n-1)!!)^2.
E.g.f.: exp(Sum_{n >= 1} b(n)*x^n/n) with b(n) = A001818(n) = ((2*n-1)!!)^2.
denom(a(n)/2^n) = A001316(n); numer(a(n)/2^n) = [1, 1, 5, 239, 8531, 2726207, ...].

A014501 Number of graphs with loops, having unlabeled (non-isolated) nodes and n labeled edges.

Original entry on oeis.org

1, 2, 7, 43, 403, 5245, 89132, 1898630, 49209846, 1517275859, 54669946851, 2269075206395, 107199678164289, 5707320919486026, 339510756324234931, 22400182888853554291, 1628654713107465602783, 129754625253841669625051

Offset: 0

Simon Plouffe, Gilbert Labelle (gilbert(AT)lacim.uqam.ca)

nonn

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

Andrew Howroyd, Table of n, a(n) for n = 0..200
G. Labelle, Counting enriched multigraphs according to the number of their edges (or arcs), Discrete Math., 217 (2000), 237-248.
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]

Row n=2 of A331161.

Cf. A020554, A020555, A014500.

E.g.f.: exp(-1+x/2)*Sum((1+x)^binomial(n+1, 2)/n!, n=0..infinity) [probably in the Labelle paper]. - Vladeta Jovovic, Apr 27 2004

A355284 Expansion of e.g.f. 1 / (1 + x + x^2/2 + log(1 - x)).

Original entry on oeis.org

1, 0, 0, 2, 6, 24, 200, 1560, 12936, 130368, 1458432, 17623440, 233922480, 3376625472, 52382131776, 870882440064, 15459372915840, 291596692838400, 5824039155720192, 122814724467223296, 2726547887891407104, 63562453551393223680, 1552499303360183700480

Offset: 0

Ilya Gutkovskiy, Jun 26 2022

nonn

Cf. A007840, A038205, A102233, A226226, A355285.

nmax = 22; CoefficientList[Series[1/(1 + x + x^2/2 + Log[1 - x]), {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, k] (k - 1)! a[n - k], {k, 3, n}]; Table[a[n], {n, 0, 22}]

my(x='x+O('x^30)); Vec(serlaplace(1/(1 + x + x^2/2 + log(1 - x)))) \\ Michel Marcus, Jun 27 2022

E.g.f.: 1 / (1 - Sum_{k>=3} x^k/k).

a(0) = 1; a(n) = Sum_{k=3..n} binomial(n,k) * (k-1)! * a(n-k).

A098233 Consider the family of ordinary multigraphs. Sequence gives the triangle read by rows giving coefficients of polynomials arising from enumeration of those multigraphs on n edges.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 4, 7, 3, 1, 1, 13, 46, 47, 25, 6, 1, 1, 40, 295, 587, 516, 235, 65, 10, 1, 1, 121, 1846, 6715, 9690, 7053, 3006, 800, 140, 15, 1, 1, 364, 11347, 73003, 170051, 189458, 119211, 46795, 12201, 2170, 266, 21, 1, 1, 1093, 68986, 768747

Offset: 0

N. J. A. Sloane, Oct 26 2004

Also gives number T(n, k) of partitions of the multiset {1, 1, 2, 2, ..., n, n} into k nonempty subsets, for 2 <= k <= 2n. - Marko Riedel, Jan 22 2023

			The first few polynomials are:
  1,
  x^2,
  x^2+x^3+x^4,
  x^2+4x^3+7x^4+3x^5+x^6,
  x^2+13x^3+46x^4+47x^5+25x^6+6x^7+x^8,
  x^2+40x^3+295x^4+587x^5+516x^6+235x^7+65x^8+10x^9+x^10,
  ...
Triangle starts:
  1;
  1;
  1,  1,   1;
  1,  4,   7,   3,   1;
  1, 13,  46,  47,  25,   6,  1;
  1, 40, 295, 587, 516, 235, 65, 10, 1;
  ...

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

Steve Butler, Fan Chung, Jay Cummings, and R. L. Graham, Juggling card sequences, arXiv:1504.01426 [math.CO], 2015.
L. Comtet, Birecouvrements et birevêtements d'un ensemble fini, Studia Sci. Math. Hungar 3 (1968): 137-152. [Annotated scanned copy. Warning: the table of v(n,k) has errors.]
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]

Cf. A360037, A360038, A360039, A020554 (row sums).

A098362 Consider the family of multigraphs enriched by the species of partitions. Sequence gives the triangle read by rows giving coefficients of polynomials arising from enumeration of those multigraphs on n arcs of 7 different colors.

Original entry on oeis.org

1, 7, 105, 196, 49, 1673, 11564, 13181, 4116, 343, 28133, 566636, 1643999, 1407672, 453446, 57624, 2401, 496769, 26784380, 168588665, 298710468, 207080419, 65115120, 9772070, 672280, 16807

Offset: 0

N. J. A. Sloane, Oct 26 2004

			1, 7x^2, 105x^2+196x^3+49x^4, 1673x^2+11564x^3+13181x^4+4116x^5+343x^6, 28133x^2+566636x^3+1643999x^4+1407672x^5+453446x^6+57624x^7+2401x^8, ...

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]

A098621 Consider the family of multigraphs enriched by the species of partitions. Sequence gives number of those multigraphs with n loops and edges.

Original entry on oeis.org

1, 2, 11, 95, 1173, 19364, 407447, 10552664, 327719713, 119600021230, 504784756672, 24321086122048

Offset: 0

N. J. A. Sloane, Oct 26 2004

nonn

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]

A098624 Consider the family of multigraphs enriched by the species of derangements. Sequence gives number of those multigraphs with n labeled edges.

Original entry on oeis.org

1, 0, 1, 2, 15, 84, 750, 6852, 79639, 1006184, 14875218, 241078100, 4392257716, 87279581232, 1905609327583, 45008114794874, 1150897256534370, 31580332783936416, 928535967078634497, 29090873853321687666, 969132936087009709174, 34198721664081728281400

Offset: 0

N. J. A. Sloane, Oct 26 2004

nonn

G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004.

Andrew Howroyd, Table of n, a(n) for n = 0..200
G. Paquin, Dénombrement de multigraphes enrichis, Mémoire, Math. Dept., Univ. Québec à Montréal, 2004. [Cached copy, with permission]

Cf. A000166, A014500, A098620, A098625, A098626, A098627.

\\ R(n) is A000166 as e.g.f.; EnrichedGnSeq defined in A098620.
R(n)={exp(-x + O(x*x^n))/(1-x)}
EnrichedGnSeq(R(20)) \\ Andrew Howroyd, Jan 12 2021

E.g.f.: B(R(x)) where B(x) is the e.g.f. of A014500 and 1 + R(x) is the e.g.f. of A000166. - Andrew Howroyd, Jan 12 2021

Terms a(14) and beyond from Andrew Howroyd, Jan 12 2021

cp's OEIS Frontend

A098623 Consider the family of directed multigraphs enriched by the species of set partitions. Sequence gives number of those multigraphs with n labeled arcs.

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PARI

Formula

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A014505 Number of digraphs with unlabeled (non-isolated) nodes and n labeled edges.

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Author

Keywords

References

Links

Crossrefs

Formula

A047865 Number of derangements of n where minimal cycle size is at least 4.

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Author

Keywords

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A274760 The multinomial transform of A001818(n) = ((2*n-1)!!)^2.

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Maple

Mathematica

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A014501 Number of graphs with loops, having unlabeled (non-isolated) nodes and n labeled edges.

Views

Author

Keywords

References

Links

Crossrefs

Formula

A355284 Expansion of e.g.f. 1 / (1 + x + x^2/2 + log(1 - x)).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

PARI

Formula

A098233 Consider the family of ordinary multigraphs. Sequence gives the triangle read by rows giving coefficients of polynomials arising from enumeration of those multigraphs on n edges.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A098362 Consider the family of multigraphs enriched by the species of partitions. Sequence gives the triangle read by rows giving coefficients of polynomials arising from enumeration of those multigraphs on n arcs of 7 different colors.

Views

Author

Keywords

Examples

References

Links

A098621 Consider the family of multigraphs enriched by the species of partitions. Sequence gives number of those multigraphs with n loops and edges.

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