A104209 -id:A104209 - cp's OEIS frontend

A173217 G.f.: A(x) = Sum_{n>=0} (1 + x)^(n^2) / 2^(n+1).

1, 3, 36, 744, 21606, 807912, 36948912, 1997801520, 124666314300, 8817945612300, 697162848757056, 60925366551278592, 5831682410241684192, 606763511537812563648, 68184018356901256320192, 8229830886505821175612416, 1061871008421711265790015880, 145851902823090076435152800208, 21247730059665104564252809209792

Offset: 0

Paul D. Hanna, Mar 05 2010

nonn

Variant of A104209, which enumerates labeled directed multigraphs.

Number of labeled digraphs with n edges and no vertices of degree zero, in which loops are permitted but not duplicate edges. - David Bevan, Apr 22 2013

Paul D. Hanna, Table of n, a(n) for n = 0..100
StackExchange, Combinatorial puzzle (2013)

Cf. A104209, A173218, A301466, A301468, A265936.

Table[Sum[Binomial[k^2, n]/2^(k+1), {k, 0, Infinity}], {n, 0, 20}] (* Vaclav Kotesovec, Mar 21 2018 *)
Table[Sum[StirlingS1[n, j] * HurwitzLerchPhi[1/2, -2*j, 0]/2, {j, 0, n}] / n!, {n, 0, 20}] (* Vaclav Kotesovec, Mar 21 2018 *)

{a(n)=local(A=sum(m=0,n^2+100,(1+x +O(x^(n+2)))^(m^2)/2^(m+1)));round(polcoeff(A,n))}

/* Continued fraction expression: */
{a(n) = my(CF=1, q = 1+x +x*O(x^n)); for(k=0, n, CF = 1/(2 - q^(4*n-4*k+1)/(1 - q^(2*n-2*k+1)*(q^(2*n-2*k+2) - 1)*CF)) ); polcoeff(CF, n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 18 2018

G.f.: 1/(2 - q/(1 - q*(q^2-1)/(2 - q^5/(1 - q^3*(q^4-1)/(2 - q^9/(1 - q^5*(q^6-1)/(2 - q^13/(1 - q^7*(q^8-1)/(2 - ...))))))))) where q = (1+x), a continued fraction due to a partial elliptic theta function identity. - Paul D. Hanna, Mar 18 2018
G.f.: Sum_{n>=0} 1/2^(n+1) * (1+x)^n * Product_{k=1..n} (2 - (1+x)^(4*k-3)) / (2 - (1+x)^(4*k-1)), due to a q-series identity. - Paul D. Hanna, Mar 18 2018
a(n) ~ 2^(2*n - 1/2 - log(2)/8) * n^n / (exp(n) * log(2)^(2*n + 1)). - Vaclav Kotesovec, Mar 21 2018

A052171 Number of directed multigraphs with loops on an infinite set of nodes containing a total of n arcs.

Original entry on oeis.org

1, 2, 11, 52, 296, 1724, 11060, 74527, 533046, 3999187, 31412182, 257150093, 2188063401, 19299062896, 176059781439, 1657961491087, 16089088019098, 160643776819423, 1648068916722737, 17351137043998280, 187255329043638437, 2069426416836401375, 23397468305569068113, 270406562951254606048, 3191908298072118225550, 38454691427657997701136

Offset: 0

Vladeta Jovovic, Jan 26 2000

nonn

Row sums of A136564, limiting values of A138107. - Benoit Jubin, May 13 2008

Euler transform of A137975. - M. F. Hasler, Jul 31 2017

Andrew Howroyd, Table of n, a(n) for n = 0..50
Banglei Guan, Ji Zhao, and Laurent Kneip, Six-Point Method for Multi-Camera Systems with Reduced Solution Space, arXiv:2402.18066 [cs.CV], 2024. See p. 10.
J.-C. Novelli, J.-Y. Thibon and N. M. Thiery, Algèbres de Hopf de graphes, C.R. Acad. Sci. Paris (Comptes Rendus Mathématique), 339 (2004), 607-610.
Sanjaye Ramgoolam, Permutation Invariant Gaussian Matrix Models, arXiv:1809.07559 [hep-th], 2018.

Cf. A050535, A007717, A005993, A050927, A050929.

Cf. A104209. Cf. A137975 (connected).

a(n) = A138107(2*n,n). - Max Alekseyev, Oct 17 2017

a(16)-a(25) from Max Alekseyev, Jun 21 2011

A122420 Number of labeled directed multigraphs with n arcs and with no vertex of indegree 0.

Original entry on oeis.org

1, 0, 1, 10, 120, 1778, 31685, 661940, 15882128, 430607370, 13022755068, 434697574538, 15875944361864, 629756003982336, 26963278837704185, 1239382820431888898, 60875147436141987437, 3181961834442383306068

Offset: 0

Vladeta Jovovic, Sep 03 2006

Cf. A104209.

A122418 := proc(n) option remember ; add( combinat[stirling2](n,k)*(k-1)^n*k!,k=0..n) ; end: A122420 := proc(n) option remember ; add( abs(combinat[stirling1](n,k))*A122418(k),k=0..n)/n! ; end: for n from 0 to 30 do printf("%d, ",A122420(n)) ; od ; # R. J. Mathar, May 18 2007

Table[1/n!*Sum[Abs[StirlingS1[n,k]]*Sum[(m-1)^k*m!*StirlingS2[k,m],{m,0,k}],{k,0,n}],{n,0,20}] (* Vaclav Kotesovec, May 07 2014 *)

a(n) = (1/n!)*Sum_{k=0..n} |Stirling1(n,k)|*A122418(k). G.f.: A(x/(1-x)) where A(x) is g.f. for A122419.

a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.1221803955695846906452721220983425... . - Vaclav Kotesovec, May 07 2014

More terms from R. J. Mathar, May 18 2007

A121316 Unlabeled version of A055203: number of different relations between n intervals (of nonzero length) on a line, up to permutation of intervals.

Original entry on oeis.org

1, 1, 7, 75, 1105, 20821, 478439, 12977815, 405909913, 14382249193, 569377926495, 24908595049347, 1193272108866953, 62128556769033261, 3493232664307133871, 210943871609662171055, 13615857409567572389361, 935523911378273899335537

Offset: 0

Goran Kilibarda and Vladeta Jovovic, Aug 25 2006

nonn

Also number of labeled multigraphs without isolated vertices and with n edges.

Nathaniel Johnston, Table of n, a(n) for n = 0..125
A. N. Bhavale, B. N. Waphare, Basic retracts and counting of lattices, Australasian J. of Combinatorics (2020) Vol. 78, No. 1, 73-99.

Row n=2 of A330942.

Cf. A055203, A121251, A104209.

seq(sum(binomial(k*(k-1)/2+n-1,n)/2^(k+1),k=0..infinity),n=0..20);
with(combinat): A121316:=proc(n) return (1/n!)*add(abs(stirling1(n,k))*A055203(k),k=0..n): end: seq(A121316(n),n=0..20); # Nathaniel Johnston, Apr 28 2011

Table[Sum[Binomial[k*(k-1)/2+n-1,n]/2^(k+1),{k,0,Infinity}],{n,0,20}] (* Vaclav Kotesovec, Mar 15 2014 *)

a(n) = {sum(j=0, 2*n, binomial(binomial(j,2)+n-1, n) * sum(i=j, 2*n, (-1)^(i-j)*binomial(i,j)))} \\ Andrew Howroyd, Feb 09 2020

a(n) = (1/n!)* Sum_{k=0..n} |Stirling1(n,k)|*A055203(k).
a(n) = Sum_{k>=0} binomial(k*(k-1)/2+n-1,n)/2^(k+1).
a(n) ~ n^n * 2^(n-1 + log(2)/4) / (exp(n) * (log(2))^(2*n+1)). - Vaclav Kotesovec, Mar 15 2014
a(n) = Sum_{j=0..2*n} binomial(binomial(j,2)+n-1, n) * (Sum_{i=j..2*n} (-1)^(i-j)*binomial(i,j)). - Andrew Howroyd, Feb 09 2020

A301311 G.f.: Sum_{n>=0} 2^n * (1-x)^(-n^2) / 3^(n+1).

Original entry on oeis.org

1, 10, 370, 22570, 1924270, 210821290, 28223418010, 4464779024650, 814901395935550, 168556843188104050, 38965275697707264970, 9955529477371346769010, 2785811940289987110605590, 847316256984037311888049090, 278329013908504193489288029090, 98197864581209379156337136722690, 37034491818759647215732974465421990, 14868275488492647637389364332301206490

Offset: 0

Paul D. Hanna, Mar 18 2018

nonn

Is there a finite expression for the terms of this sequence?

a(n) is divisible by 10 for n>0 (conjecture).

			G.f.: A(x) = 1 + 10*x + 370*x^2 + 22570*x^3 + 1924270*x^4 + 210821290*x^5 + 28223418010*x^6 + 4464779024650*x^7 + 814901395935550*x^8 + ...
such that
A(x) = 1/3 + 2/(1-x)/3^2 + 2^2/(1-x)^4/3^3 + 2^3/(1-x)^9/3^4 + 2^4/(1-x)^16/3^5 + 2^5/(1-x)^25/3^6 + 2^6/(1-x)^36/3^7 + 2^7/(1-x)^49/3^8  + 2^8/(1-x)^64/3^9 + ...

Paul D. Hanna, Table of n, a(n) for n = 0..100

Cf. A104209, A301310.

/* Continued fraction expression: */
{a(n) = my(CF=1, q = 1/(1-x +x*O(x^n))); for(k=0, n, CF = 1/(3 - 2*q^(4*n-4*k+1)/(1 - 2*q^(2*n-2*k+1)*(q^(2*n-2*k+2) - 1)*CF)) ); polcoeff(CF, n)}
for(n=0, 30, print1(a(n), ", "))

G.f.: 1/(3 - 2*q/(1 - 2*q*(q^2-1)/(3 - 2*q^5/(1 - 2*q^3*(q^4-1)/(3 - 2*q^9/(1 - 2*q^5*(q^6-1)/(3 - 2*q^13/(1 - 2*q^7*(q^8-1)/(3 - ...))))))))) where q = 1/(1-x), a continued fraction due to a partial elliptic theta function identity.
G.f.: Sum_{n>=0} 2^n/3^(n+1) * (1-x)^n * Product_{k=1..n} (3*(1-x)^(4*k-3) - 2) / (3*(1-x)^(4*k-1) - 2), due to a q-series identity.
a(n) = Sum_{k>=0} 2^k * binomial(k^2 + n-1, n) / 3^(k+1).
a(n) ~ 2^(2*n + 1/2 - log(3/2)/8) * 3^(log(3/2)/8 - 1) * n^n / (exp(n) * (log(3/2))^(2*n + 1)). - Vaclav Kotesovec, Mar 21 2018

A121886 a(n) = (1/n!)* Sum_{k=0..n} |Stirling1(n,k)|*A122399(k).

Original entry on oeis.org

1, 1, 5, 40, 444, 6324, 110023, 2261576, 53632424, 1441341350, 43290170494, 1437020742408, 52243864528990, 2064488610832106, 88106523694973953, 4038627301344466648, 197888243609535940091, 10321811633042512528240

Offset: 0

Vladeta Jovovic, Aug 31 2006

Number of square matrices with nonnegative integer entries and without zero rows such that sum of all entries is equal to n. - Vladeta Jovovic, Mar 04 2008

			G.f.: A(x) = 1 + x + 5*x^2 + 40*x^3 + 444*x^4 + 6324*x^5 +...
where
A(x) = 1 + (1/(1-x) - 1) + (1/(1-x)^2 - 1)^2 + (1/(1-x)^3 - 1)^3 + ...
Also,
A(x) = 1/2 + (1-x)/(1 + (1-x))^2 + (1-x)^2/(1 + (1-x)^2)^3 +  + (1-x)^3/(1 + (1-x)^3)^4 + (1-x)^4/(1 + (1-x)^4)^5 + ...

Cf. A104209, A220352.

Flatten[{1,Table[1/n!* Sum[Abs[StirlingS1[n,k]]*Sum[m^k*m!*StirlingS2[k, m], {m, 1, k}],{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, May 07 2014 *)

{a(n)=polcoeff(sum(m=0,n,(1/(1-x+x*O(x^n))^m-1)^m),n)}

G.f.: Sum_{n>=0} ( 1/(1-x)^n - 1 )^n.
G.f.: Sum_{n>=0} (1-x)^n / (1 + (1-x)^n)^(n+1). - Paul D. Hanna, Sep 07 2018
a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = (1+exp(1/r))*r^2 = 3.161088653865428813830172202588132491726382774188556341627278..., r = 0.8737024332396683304965683047207192982139922672025395099... is the root of the equation exp(1/r)/r + (1+exp(1/r))*LambertW(-exp(-1/r)/r) = 0, and c = 0.38377369607518184186200387319561108... . - Vaclav Kotesovec, May 07 2014

More terms from Max Alekseyev, Feb 01 2007

A120945 Triangle T(n,k) of number of labeled directed multigraphs (with loops), without isolated vertices, with n arrows and k vertices (n = 1,2,.., k = 1..2*n).

Original entry on oeis.org

1, 2, 1, 8, 18, 12, 1, 18, 108, 272, 300, 120, 1, 33, 393, 2102, 5700, 8160, 5880, 1680, 1, 54, 1122, 10688, 53550, 153132, 258720, 255360, 136080, 30240, 1, 82, 2754, 42752, 351650, 1688892, 5025832, 9540272, 11566800, 8668800, 3659040, 665280

Offset: 1

Vladeta Jovovic, Aug 19 2006

			[1,2], [1,8,18,12], [1,18,108,272,300,120], [1,33,393,2102,5700,8160,5880,1680], ....

Row sums give A104209.

T(n,k) = Sum_{j=0..k} (-1)^(k-j)*binomial(k,j)*binomial(j^2+n-1,n). Row sums give A104209.

E.g.f.: exp(-x)*Sum((1-y)^(-n^2)*x^n/n!,n=0..infinity). - Vladeta Jovovic, Aug 24 2006

A121063 Number of labeled directed multigraphs with n arcs for which every vertex has in-degree at least one and out-degree at least one.

Original entry on oeis.org

1, 0, 1, 4, 27, 246, 2783, 37424, 582153, 10276452, 202894801, 4429522252, 105943672079, 2754788353526, 77371821493913, 2334279549290960, 75286455363538607, 2584971423426768872, 94138234184851584599, 3624294240897948371036, 147080227272202880297669

Offset: 0

Vladeta Jovovic, Sep 06 2006

Nathaniel Johnston, Table of n, a(n) for n = 0..60

Cf. A052170, A056078, A104209, A121933.

n:=20: t:=taylor(sum(sum((-1)^(s-k)*binomial(s,k)*((1+x/(1-x))^(k-1)-1)^k*((1+x/(1-x))^k-1)^(s-k),k=0..s),s=0..n),x,n+1): seq(coeff(t,x,s),s=0..n); # Nathaniel Johnston, Apr 28 2011

G.f.: A(x/(1-x)) where A(x) is g.f. for A121933.

A121137 Number of labeled directed multigraphs (without loops) with n arcs and no vertex of degree 0.

Original entry on oeis.org

1, 2, 27, 572, 16787, 631362, 28980861, 1570956872, 98212870233, 6956704585554, 550626446263423, 48163137319172436, 4613554511554200251, 480324019903607680066, 54004504167811544647161, 6521368218660772789452944, 841771274136198763040518633

Offset: 0

Vladeta Jovovic, Sep 06 2006

Nathaniel Johnston, Table of n, a(n) for n = 0..50
Math StackExchange, Marko Riedel et al., Labeled multigraphs
Marko Riedel, Proof of closed form by Egorychev method.

Cf. A052170 (unlabeled analog), A104209, A052171.

seq(sum(binomial(m*(m-1)+n-1,n)/2^(m+1),m=0..infinity),n=0..10);
# alternate program
A121137:= n -> add(add(binomial(m, q)*(-1)^(m-q)*binomial(n+q*(q-1)-1, n), q=0..m), m=0..2*n):
seq(A121137(n), n=0..20); # Marko Riedel, Jan 26 2025

a(n) = Sum_{m>=0} binomial(m*(m-1)+n-1,n)/2^(m+1).

a(n) = Sum_{m=0..2n} Sum_{q=0..m} binomial(m,q)*(-1)^(m-q)*binomial(n+q*(q-1)-1,n). - Marko Riedel, Jan 26 2025

cp's OEIS Frontend

A173217 G.f.: A(x) = Sum_{n>=0} (1 + x)^(n^2) / 2^(n+1).

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A052171 Number of directed multigraphs with loops on an infinite set of nodes containing a total of n arcs.

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A122420 Number of labeled directed multigraphs with n arcs and with no vertex of indegree 0.

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A121316 Unlabeled version of A055203: number of different relations between n intervals (of nonzero length) on a line, up to permutation of intervals.

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A301311 G.f.: Sum_{n>=0} 2^n * (1-x)^(-n^2) / 3^(n+1).

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A121886 a(n) = (1/n!)* Sum_{k=0..n} |Stirling1(n,k)|*A122399(k).

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A120945 Triangle T(n,k) of number of labeled directed multigraphs (with loops), without isolated vertices, with n arrows and k vertices (n = 1,2,.., k = 1..2*n).

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A121063 Number of labeled directed multigraphs with n arcs for which every vertex has in-degree at least one and out-degree at least one.

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