A020556 -id:A020556 - cp's OEIS frontend

A020558 Number of ordered multigraphs on n labeled edges (without loops).

1, 1, 4, 27, 274, 3874, 71995, 1682448, 47840813, 1615315141, 63566760077, 2873099980637, 147384910116793, 8496500896980637, 545845612016485842, 38797966029876716897, 3032005571734589578076

Offset: 0

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

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]

E.g.f.: exp((3*x-2)/(2-2*x))*Sum(1/(n!*(1-x)^binomial(n, 2)), n = 0 .. infinity). a(n) = Sum((-1)^(n-k)*Stirling1(n, k)*A020554(k), k=0..n). - Vladeta Jovovic, May 02 2004

E.g.f.: exp(x/(2-2*x))*Sum(A020556(n)*(-log(1-x)/2)^n/n!, n=0..infinity). - Vladeta Jovovic, May 02 2004

A078738 Generalized Bell numbers B_{3,2}(n).

Original entry on oeis.org

1, 13, 355, 16333, 1121881, 106708921, 13354028563, 2118817455385, 414426460442833, 97746679844312581, 27311169061720393411, 8908525371578726747173, 3350963996380181114090665, 1438463413778071631322236593, 698374517715612292764726380851

Offset: 1

N. J. A. Sloane, Dec 21 2002

Vaclav Kotesovec, Table of n, a(n) for n = 1..240
P. Blasiak, K. A. Penson and A. I. Solomon, The Boson Normal Ordering Problem and Generalized Bell Numbers, arXiv:quant-ph/0212072, 2002.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, arXiv:quant-ph/0402027, 2004.
P. Blasiak, K. A. Penson and A. I. Solomon, The general boson normal ordering problem, Phys. Lett. A 309 (2003) 198-205.
M. Schork, On the combinatorics of normal ordering bosonic operators and deforming it, J. Phys. A 36 (2003) 4651-4665.

B_{1, 1} = A000110, B_{2, 1} = A000262, B_{3, 1} = A020556 and B_{3, 3} = A069223. Row sums of A078740.

Alternating row sums A090437.

a[n_] := (n+1)*n!^2*Sum[(-1)^k*HypergeometricPFQ[{2-k, n+1, n+2}, {2, 3}, 1]/(2*(k-2)!), {k, 2, 2n}]; Array[a, 13] (* Jean-François Alcover, Sep 01 2015 *)
Table[Sum[(n + k)!*(n + k + 1)!/(k!*(k + 1)!*(k + 2)!), {k, 0, Infinity}]/E, {n, 1, 20}] (* Vaclav Kotesovec, Jul 27 2018 *)

nmax = 20; p = floor(3*nmax*log(nmax)); default(realprecision, p);
for(n=1, nmax, print1(round(exp(-1)*suminf(k=0, (n+k)!*(n+k+1)!/(k!*(k+1)!*(k+2)!))), ", ")) \\ G. C. Greubel and Vaclav Kotesovec, Jul 28 2018

a(n) = Sum_{k=2..2*n} A078740(n, k) = Sum_{k=1..infinity} (1/k!)*Product_{j=1..n}(fallfac(k+(j-1)*(3-2), 2))/exp(1), n>=1. From eq.(9) of the Blasiak et al. reference with r=3, s=2. fallfac(n, m) := A008279(n, m) (falling factorials triangle). a(0) := 1 may be added.
a(n) = Sum_{k>=0} ((n+k)!*(n+k+1)!/(k!*(k+1)!*(k+2)!))/exp(1), n>=1. From eq.(40) of the Blasiak et al. reference. [corrected by Vaclav Kotesovec, Jul 27 2018]
E.g.f. for a(n)/n! with a(0)=(exp(1)-1)/exp(1) added: Sum_{k>=0} (hypergeom([k+2, k+1], [1], z)/(k+2)!)/exp(1). From eq. (41) of the Blasiak et al. reference.

Edited by Wolfdieter Lang, Dec 23 2003

A157280 a(n) arises in the normal ordering of n-th power of the operator (d/dx)(x(d/dx))^4.

Original entry on oeis.org

1, 52, 43833, 149670844, 1346634725665, 25571928251231076, 893591647147188285577, 52327970757667659912764908, 4796836032234830356783078467969

Offset: 1

Karol A. Penson, Feb 26 2009

nonn

this sequence generates the fifth terms of the following sequences:
a(2)=52=A000110(5), a(3)=43833=A020556(5), a(4)=149670844=A069223(5),
a(5)=1346634725665=A071379(5),a(6)=25571928251231076=A070227(5)

nMax = 8; kMax = 50; seq0 = {}; seq = {1}; While[seq != seq0, seq0 = seq; seq = (1/E Sum[HypergeometricPFQ[{k+1, k+1, k+1, k+1}, {1, 1, 1, 1}, x]/k!, {k, 0, kMax}] + O[x]^(nMax+1) // CoefficientList[#, x]&) Range[0, nMax]!^5 // Round; kMax += 10; Print[kMax]]; A157280 = seq (* Jean-François Alcover, Nov 07 2016 *)

Sequence defined through the following hypergeometric-type generating function, in Maple notation:
exp(-1)*sum(hypergeom([k+1,k+1,k+1,k+1],[1,1,1,1],x)/k!,k=0..infinity)=sum(a(n)*x^n/(n!)^5,n=0..infinity),
which is itself an infinite sum of hypergeometric functions.

A340823 a(n) = exp(-1) * Sum_{k>=0} (k*(k - n))^n / k!.

Original entry on oeis.org

1, 1, 3, 5, 124, -2075, 91993, -4709903, 312334595, -25531783799, 2524083665172, -296260739274275, 40667620527027177, -6446882734412545043, 1167717545574222779643, -239452569059443831797303, 55146244227862697483251020, -14163492441645773105212592623

Offset: 0

Ilya Gutkovskiy, Jan 22 2021

sign

G. C. Greubel, Table of n, a(n) for n = 0..260

Cf. A000110, A020556, A020557, A290219, A334243, A340822.

A340823:= func< n | (&+[(-n)^j*Binomial(n,j)*Bell(2*n-j): j in [0..n]]) >;
[A340823(n): n in [0..30]]; // G. C. Greubel, Jun 12 2024

Table[Exp[-1] Sum[(k (k - n))^n/k!, {k, 0, Infinity}], {n, 0, 17}]
Join[{1}, Table[Sum[Binomial[n, k] BellB[2 n - k] (-n)^k, {k, 0, n}], {n, 1, 17}]]

def A340823(n): return sum( binomial(n,k)*bell_number(2*n-k)*(-n)^k for k in range(n+1))
[A340823(n) for n in range(31)] # G. C. Greubel, Jun 12 2024

a(n) = Sum_{k=0..n} binomial(n,k) * Bell(2*n-k) * (-n)^k.

A132219 Number of line graphs on n labeled nodes.

Original entry on oeis.org

1, 2, 8, 66, 774, 11885, 230858, 5512821, 157926273, 5323010347, 207833183972, 9280406921629, 468879370270319, 26561064972285450, 1673867356632729422, 116555375609273658636, 8914259769349445083676, 744876885721366948399627

Offset: 1

Vladeta Jovovic, Aug 14 2007

nonn

Cf. A132220 (unlabeled case).

E.g.f.: exp(x/2-x^3/6)*Sum(A020556(n)*(log(1+x)/2)^n/n!, n=0..infinity).

A340837 a(n) = (1/2) * Sum_{k>=0} (k*(k - 1))^n / 2^k.

Original entry on oeis.org

1, 2, 52, 3272, 382672, 71819552, 19755648832, 7489898916992, 3743721038908672, 2385494267756237312, 1887436919680269939712, 1815491288416066631616512, 2086364959404184854563049472, 2823211429546048668686123343872, 4443155724532239407325655263035392

Offset: 0

Ilya Gutkovskiy, Jan 23 2021

nonn

Cf. A000670, A020556, A052841, A080163, A098696, A122101, A249938.

Table[(1/2) Sum[(k (k - 1))^n/2^k, {k, 0, Infinity}], {n, 0, 14}]
Table[(1/2) Sum[(-1)^k Binomial[n, k] HurwitzLerchPhi[1/2, k - 2 n, 0], {k, 0, n}], {n, 0, 14}]

a(n) = Sum_{k=0..n} (-1)^k * binomial(n,k) * A000670(2*n-k).
a(n) = 2 * A080163(n) for n > 0. - Hugo Pfoertner, Jan 23 2021
a(n) = A122101(2*n,n). - Alois P. Heinz, Jun 23 2023

A020560 Number of ordered oriented multigraphs on n labeled arcs (without loops).

Original entry on oeis.org

1, 1, 8, 110, 2262, 63822, 2335856, 106643980, 5901564892, 387251339836, 29612207398688, 2601877956513640, 259646660158526856, 29140784853464966632, 3647614440065070983488, 505553321559685235600720, 77096697572750991426551056, 12864789849511242132008021520

Offset: 0

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

nonn

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. A020556.

a(n) = Sum_{k=0..n} (-1)^(n-k) * Stirling1(n, k) * A020556(k). - Sean A. Irvine, Apr 24 2019

A308538 a(n) = exp(1) * Sum_{k>=0} (-1)^k(k(k + 1))^n/k!.

Original entry on oeis.org

1, -1, 3, -11, -47, 2111, -18589, -671803, 24578625, 328227551, -44122149981, -17436709707, 129678315723089, -1671362093527073, -621466693739336509, 16245508786340273349, 4705502566521408713473, -172007688705762528111809, -54084940569279187064459581

Offset: 0

Ilya Gutkovskiy, Jun 06 2019

sign

Cf. A000587, A020556, A090211, A094577.

Table[Exp[1] Sum[(-1)^k (k (k + 1))^n/k!, {k, 0, Infinity}], {n, 0, 18}]
Table[Sum[Binomial[n, k] BellB[2 n - k, -1], {k, 0, n}], {n, 0, 18}]

a(n) = Sum_{k=0..n} binomial(n,k)*A000587(2*n-k).

cp's OEIS Frontend

A020558 Number of ordered multigraphs on n labeled edges (without loops).

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A078738 Generalized Bell numbers B_{3,2}(n).

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A157280 a(n) arises in the normal ordering of n-th power of the operator (d/dx)(x(d/dx))^4.

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A340823 a(n) = exp(-1) * Sum_{k>=0} (k*(k - n))^n / k!.

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A132219 Number of line graphs on n labeled nodes.

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A340837 a(n) = (1/2) * Sum_{k>=0} (k*(k - 1))^n / 2^k.

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Mathematica

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A020560 Number of ordered oriented multigraphs on n labeled arcs (without loops).

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A308538 a(n) = exp(1) * Sum_{k>=0} (-1)^k*(k*(k + 1))^n/k!.

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Programs

Mathematica

Formula

A308538 a(n) = exp(1) * Sum_{k>=0} (-1)^k(k(k + 1))^n/k!.