A097629 -id:A097629 - cp's OEIS frontend

A062735 Triangular array T(n,k) giving number of weakly connected digraphs with n labeled nodes and k arcs (n >= 1, 0 <= k <= n(n-1)).

1, 0, 2, 1, 0, 0, 12, 20, 15, 6, 1, 0, 0, 0, 128, 432, 768, 920, 792, 495, 220, 66, 12, 1, 0, 0, 0, 0, 2000, 11104, 33880, 73480, 123485, 166860, 184426, 167900, 125965, 77520, 38760, 15504, 4845, 1140, 190, 20, 1, 0, 0, 0, 0, 0, 41472, 337920, 1536000, 5062080

Offset: 1

Vladeta Jovovic, Jul 12 2001

			1;
0, 2, 1;
0, 0, 12, 20,   15,    6,      1;
0, 0, 0, 128,  432,  768,    920,    792,    495,    220,     66,    12, 1;
0, 0, 0,   0, 2000, 11104, 33880,  73480, 123485, 166860, 184426, 167900, ...;
0, 0, 0,   0,    0, 41472, 337920,1536000,5062080,.. ;
0, 0, 0,   0,    0,     0, 1075648,...

Andrew Howroyd, Table of n, a(n) for n = 1..2680 (rows 1..20)
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO], 2017; Table 76.

Cf. A003027 (row sums), A054733 (unlabeled case), A057273 (strongly connected), A097629 (diagonal), A123554 (not necessarily connected).

nn=7;s=Sum[(1+y)^(n^2-n) x^n/n!,{n,0,nn}];Range[0,nn]!CoefficientList[Series[Log[ s]+1,{x,0,nn}],{x,y}]//Grid  (* returns triangle indexed from n = 0, Geoffrey Critzer, Oct 07 2012 *)

row(n)={Vecrev(n!*polcoef(1 + log(sum(k=0, n, (1+y)^(k*(k-1))*x^k/k!, O(x*x^n))), n))}
{ for(n=0, 5, print(row(n))) } \\ Andrew Howroyd, Jan 11 2022

E.g.f.: 1+log( Sum_{n >= 0, k >= 0} binomial(n*(n-1), k)*x^n/n!*y^k ).

A152555 Coefficients in a q-analog of the function LambertW(-2x)/(-2x), as a triangle read by rows.

Original entry on oeis.org

1, 2, 7, 5, 30, 42, 42, 14, 143, 297, 462, 495, 363, 198, 42, 728, 2002, 4004, 6006, 7436, 7436, 6292, 4290, 2288, 858, 132, 3876, 13260, 31824, 58604, 91364, 122876, 145535, 153361, 143936, 120185, 87971, 56329, 29939, 12584, 3575, 429, 21318, 87210

Offset: 0

Paul D. Hanna, Dec 07 2008

			Triangle begins:
  1;
  2;
  7,5;
  30,42,42,14;
  143,297,462,495,363,198,42;
  728,2002,4004,6006,7436,7436,6292,4290,2288,858,132;
  3876,13260,31824,58604,91364,122876,145535,153361,143936,120185,87971,56329,29939,12584,3575,429;
  21318,87210,242250,519384,945744,1508070,2165664,2826420,3392520,3756626,3853322,3662106,3221330,2613240,1944324,1313760,794614,420784,185640,64090,14586,1430;...
where row sums = 2*(2*n+2)^(n-1) (A097629).
Row sums at q=-1 = 2*(2*n+2)^[(n-1)/2] (A152556).
The generating function starts:
A(x,q) = 1 + 2*x + (7 + 5*q)*x^2/faq(2,q) + (30 + 42*q + 42*q^2 + 14*q^3)*x^3/faq(3,q) + (143 + 297*q + 462*q^2 + 495*q^3 + 363*q^4 + 198*q^5 + 42*q^6)*x^4/faq(4,q) + ...
The q-factorial of n is faq(n,q) = Product_{k=1..n} (q^k-1)/(q-1): faq(0,q)=1, faq(1,q)=1, faq(2,q)=(1+q), faq(3,q)=(1+q)*(1+q+q^2), faq(4,q)=(1+q)*(1+q+q^2)*(1+q+q^2+q^3), ...
Special cases.
q=0: A(x,0) = 1 + 2*x + 7*x^2 + 30*x^3 + 143*x^4 + 728*x^5 +... (A006013)
q=1: A(x,1) = 1 + 2*x + 12/2*x^2 + 128/6*x^3 + 2000/24*x^4 + 41472/120*x^5 +...
q=2: A(x,2) = 1 + 2*x + 17/3*x^2 + 394/21*x^3 + 21377/315*x^4 + 2537724/9765*x^5 +...
q=3: A(x,3) = 1 + 2*x + 22/4*x^2 + 912/52*x^3 + 126692/2080*x^4 + 56277344/251680*x^5 +...

Paul D. Hanna, Rows 0 to 30 of the triangle, flattened.
Eric Weisstein's World of Mathematics, q-Exponential Function.
Eric Weisstein's World of Mathematics, q-Factorial.

Cf. A097629 (row sums), A006013 (column 0), A000108 (right border), A152559.
Cf. A152556 (q=-1), A152557 (q=2), A152558 (q=3).
Cf. variants: A152290, A152550.

{T(n,k)=local(e_q=1+sum(j=1,n,x^j/prod(i=1,j,(q^i-1)/(q-1))), LW2_q=serreverse(x/(e_q+x*O(x^n))^2)/x); polcoeff(polcoeff(LW2_q+x*O(x^n),n,x)*prod(i=1,n,(q^i-1)/(q-1))+q*O(q^k),k,q)}

G.f.: A(x,q) = Sum_{n>=0} Sum_{k=0..n*(n-1)/2} T(n,k)*q^k*x^n/faq(n,q), where faq(n,q) is the q-factorial of n.
G.f.: A(x,q) = (1/x)*Series_Reversion( x/e_q(x,q)^2 ) where e_q(x,q) = Sum_{n>=0} x^n/faq(n,q) is the q-exponential function.
G.f. satisfies: A(x,q) = e_q( x*A(x,q), q)^2 and A( x/e_q(x,q)^2, q) = e_q(x,q)^2.
G.f. at q=1: A(x,1) = LambertW(-2*x)/(-2*x).
Row sums at q=+1: Sum_{k=0..n*(n-1)/2} T(n,k) = 2*(2*n+2)^(n-1).
Row sums at q=-1: Sum_{k=0..n*(n-1)/2} T(n,k)*(-1)^k = 2*(2*n+2)^[(n-1)/2].
Sum_{k=0..n*(n-1)/2} T(n,k)*exp(2*Pi*I*k/n) = 2 for n>=1; i.e., the n-th row sum at q = exp(2*Pi*I/n), the n-th root of unity, equals 2 for n>=1. - Vladeta Jovovic
Sum_{k=0..binomial(n,2)} T(n,k)*q^k = Sum_{pi} 2*(2*n+1)!/(2*n-k+2)!*faq(n,q)/Product_{i=1..n} e(i)!*faq(i,q)^e(i), where pi runs over all nonnegative integer solutions to e(1)+2*e(2)+...+n*e(n) = n and k = e(1)+e(2)+...+e(n). - Vladeta Jovovic, Dec 07 2008

A097627 Number of rooted directed trees on n nodes with a red root.

Original entry on oeis.org

1, 2, 21, 280, 5465, 134556, 4051453, 143810416, 5884797969, 272701388980, 14116335883661, 807328717090248, 50554260752606377, 3440140092356781100, 252777268861251990045, 19946622760623381708256

Offset: 1

Ralf Stephan, Aug 17 2004

nonn

Ditrees are well-colored directed trees. Well-colored means, each green vertex has at least a red child, each red vertex has no red child.

Vincenzo Librandi, Table of n, a(n) for n = 1..100
C. Banderier, J.-M. Le Bars and V. Ravelomanana, Generating functions for kernels of digraphs, arXiv:math/0411138 [math.CO], 2004.

Cf. A097629.

Rest[CoefficientList[Series[LambertW[-LambertW[-2*x]/2], {x, 0, 20}], x]* Range[0, 20]!] (* Vaclav Kotesovec, Oct 08 2013 *)

x='x+O('x^30); Vec(serlaplace(lambertw(-lambertw(-2*x)/2))) \\ G. C. Greubel, Nov 15 2017

a(n) = A052746(n) - A097628(n).
E.g.f.: -C(-C(2*x)/2), where C(x) is the e.g.f. of A000169.
a(n) ~ LambertW(1/2) * n^(n-1) * 2^n / (1+LambertW(1/2)). - Vaclav Kotesovec, Oct 08 2013

A350732 Triangle read by rows: T(n,k) is the number of weakly connected oriented graphs on n labeled nodes with k arcs, n >= 0, k=0..n*(n-1)/2.

Original entry on oeis.org

1, 0, 2, 0, 0, 12, 8, 0, 0, 0, 128, 240, 192, 64, 0, 0, 0, 0, 2000, 7104, 13120, 15360, 11520, 5120, 1024, 0, 0, 0, 0, 0, 41472, 234240, 729600, 1578240, 2531840, 3068928, 2795520, 1863680, 860160, 245760, 32768

Offset: 1

Andrew Howroyd, Jan 11 2022

			Triangle begins:
  [1] 1;
  [2] 0, 2;
  [3] 0, 0, 12,   8;
  [4] 0, 0,  0, 128,  240,  192,    64;
  [5] 0, 0,  0,   0, 2000, 7104, 13120, 15360, 11520, 5120, 1024;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1350 (rows 1..20)

Row sums are A054941.
The leading diagonal is A097629.
The unlabeled version is A350734.
Cf. A062735 (digraphs), A350731 (strongly connected).

row(n)={Vecrev(n!*polcoef(1 + log(sum(k=0, n, (1+2*y)^(k*(k-1)/2)*x^k/k!, O(x*x^n))), n))}
{ for(n=1, 5, print(row(n))) }

A097630 Number of unrooted directed trees on n nodes with a red root.

Original entry on oeis.org

1, 1, 7, 70, 1093, 22426, 578779, 17976302, 653866441, 27270138898, 1283303262151, 67277393090854, 3888789288662029, 245724292311198650, 16851817924083466003, 1246663922538961356766, 98960637955717312632721

Offset: 1

Ralf Stephan, Aug 17 2004

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
C. Banderier, J.-M. Le Bars and V. Ravelomanana, Generating functions for kernels of digraphs

Cf. A097629.

terms = 17;
Rest[CoefficientList[LambertW[-LambertW[-2x]/2] + O[x]^(terms+1), x]]* Range[0, terms-1]! (* Jean-François Alcover, Dec 13 2018 *)

seq(n)={Vec(serlaplace(lambertw(-lambertw(-2*x + O(x*x^n))/2)/x))} \\ Andrew Howroyd, Dec 13 2018

E.g.f.: A(x) = 2B - 2BC + C - 2B/C + C^2/2, with B = egf of A052746 and C = egf of A097627.
a(n) = (n-1)!*[x^n](LambertW(-LambertW(-2x)/2)). - Jean-François Alcover, Dec 13 2018
a(n) ~ 2^n * n^(n-2) / (1 + 1/LambertW(1/2)). - Vaclav Kotesovec, Feb 24 2019

A380646 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2*x)/(1 + x)^2 ).

Original entry on oeis.org

1, 4, 46, 932, 27568, 1080432, 52916176, 3115326496, 214470890496, 16914853191680, 1504252282653184, 148956086481767424, 16256865070022066176, 1938988214539948730368, 250943399365390735104000, 35026523834624205803491328, 5245178283068781060488298496, 838841884254236846183525646336

Offset: 0

Seiichi Manyama, Feb 06 2025

nonn

Index entries for reversions of series

Cf. A088690, A380647, A380648.
Cf. A065866, A377829.
Cf. A097629, A380828.
Cf. A377892.

nmax=18; CoefficientList[(1/x)InverseSeries[Series[x*Exp[-2*x]/(1 + x)^2 ,{x,0,nmax}]],x]Range[0,nmax-1]! (* Stefano Spezia, Feb 06 2025 *)

a(n) = 2*n!*sum(k=0, n, (2*n+2)^(k-1)*binomial(2*n+2, n-k)/k!);

E.g.f. A(x) satisfies A(x) = (1 + x*A(x))^2 * exp(2 * x * A(x)).
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A377892.
a(n) = 2 * n! * Sum_{k=0..n} (2*n+2)^(k-1) * binomial(2*n+2,n-k)/k!.

A377360 E.g.f. satisfies A(x) = ( 1 - log(1 - x*A(x)) )^2.

Original entry on oeis.org

1, 2, 12, 130, 2082, 44488, 1192964, 38557860, 1459988440, 63414711072, 3108861424032, 169829819311392, 10230860299538400, 673850170929176928, 48176129912775680160, 3715759452364764485280, 307545698210584533055488, 27190399275422185989742080, 2557448587458299889542868480

Offset: 0

Seiichi Manyama, Oct 26 2024

Index entries for reversions of series

Cf. A138013, A377361.

Cf. A097629, A367080, A376392.

nmax = 20; CoefficientList[1/x * InverseSeries[Series[x/(1 - Log[1 - x])^2, {x, 0, nmax + 1}], x], x] * Range[0, nmax]! (* Vaclav Kotesovec, Aug 27 2025 *)

a(n) = 2*(2*n+1)!*sum(k=0, n, abs(stirling(n, k, 1))/(2*n-k+2)!);

E.g.f.: B(x)^2, where B(x) is the e.g.f. of A367080.
a(n) = 2 * (2*n+1)! * Sum_{k=0..n} |Stirling1(n,k)|/(2*n-k+2)!.
E.g.f.: (1/x) * Series_Reversion( x/(1 - log(1-x))^2 ).
a(n) ~ sqrt(2) * LambertW(-1, -2*exp(-3))^n * (2 + LambertW(-1, -2*exp(-3)))^(n+2) * n^(n-1) / (exp(n) * sqrt(-1 - LambertW(-1, -2*exp(-3)))). - Vaclav Kotesovec, Aug 27 2025

A097631 Number of unrooted directed trees on n nodes with a green root.

Original entry on oeis.org

0, 1, 5, 58, 907, 19046, 496869, 15578130, 570573623, 23929861102, 1131235173433, 59529368839898, 3451899685313523, 218712237867226182, 15034642075916533997, 1114519318895861250082, 88631119148029975177327, 7526795487859400166772958, 679859967684397018073935617

Offset: 1

Ralf Stephan, Aug 17 2004

nonn

C. Banderier, J.-M. Le Bars and V. Ravelomanana, Generating functions for kernels of digraphs, arXiv:math/0411138 [math.CO], 2004.

Equals A097629(n) - A097630(n).

(* Note: Mathematica's ProductLog is the Lambert W function. *)
a[n_] := SeriesCoefficient[-ProductLog[-ProductLog[-2x]/2]/n - ProductLog[-2x] (ProductLog[-2x] + 2)/4, {x, 0, n}] n!;
Array[a, 17] (* Jean-François Alcover, Feb 24 2019 *)

a(n) ~ 2^(n-1) * n^(n-2) * (1 - LambertW(1/2)) / (1 + LambertW(1/2)). - Vaclav Kotesovec, Feb 24 2019

A350909 Triangle read by rows: T(n,k) is the number of weakly connected acyclic digraphs on n labeled nodes with k arcs, k=0..n*(n-1).

Original entry on oeis.org

1, 0, 2, 0, 0, 12, 6, 0, 0, 0, 128, 186, 108, 24, 0, 0, 0, 0, 2000, 5640, 7840, 6540, 3330, 960, 120, 0, 0, 0, 0, 0, 41472, 189480, 456720, 730830, 832370, 690300, 416160, 178230, 51480, 9000, 720, 0, 0, 0, 0, 0, 0, 1075648, 7178640, 26035800, 65339820

Offset: 1

Andrew Howroyd, Jan 29 2022

			Triangle begins:
  [1] 1;
  [2] 0, 2;
  [3] 0, 0, 12,   6;
  [4] 0, 0,  0, 128,  186,  108,   24;
  [5] 0, 0,  0,   0, 2000, 5640, 7840, 6540, 3330, 960, 120;
  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1350 (rows 1..20)

Row sums are A082402.
Leading diagonal is A097629.
The unlabeled version is A350449.
Cf. A057273, A062735, A081064.

G(n)={my(v=vector(n+1)); v[1]=1; for(n=1, n, v[n+1]=sum(k=1, n, -(-1)^k*(1+y)^(k*(n-k))*v[n-k+1]/k!))/n!; Ser(v)}
row(n)={Vecrev(n!*polcoef(log(G(n)), n))}
{ for(n=1, 6, print(row(n))) }

A384718 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where column k is the expansion of e.g.f. B(x)^k, where B(x) is the e.g.f. of A052750.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 5, 0, 1, 3, 12, 49, 0, 1, 4, 21, 128, 729, 0, 1, 5, 32, 243, 2000, 14641, 0, 1, 6, 45, 400, 3993, 41472, 371293, 0, 1, 7, 60, 605, 6912, 85683, 1075648, 11390625, 0, 1, 8, 77, 864, 10985, 153664, 2278125, 33554432, 410338673, 0

Offset: 0

Seiichi Manyama, Jun 08 2025

			Square array begins:
  1,     1,     1,     1,      1,      1, ...
  0,     1,     2,     3,      4,      5, ...
  0,     5,    12,    21,     32,     45, ...
  0,    49,   128,   243,    400,    605, ...
  0,   729,  2000,  3993,   6912,  10985, ...
  0, 14641, 41472, 85683, 153664, 253125, ...

Columns k=0..2 give A000007, A052750, A097629(n+1).

Cf. A058127, A232006, A384692.

PARI
```
a(n, k) = if(n==0, 1, k*(2*n+k)^(n-1));
```

A(n,k) = k * (2*n+k)^(n-1) for n > 0.

cp's OEIS Frontend

A062735 Triangular array T(n,k) giving number of weakly connected digraphs with n labeled nodes and k arcs (n >= 1, 0 <= k <= n(n-1)).

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A152555 Coefficients in a q-analog of the function LambertW(-2*x)/(-2*x), as a triangle read by rows.

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A097627 Number of rooted directed trees on n nodes with a red root.

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A350732 Triangle read by rows: T(n,k) is the number of weakly connected oriented graphs on n labeled nodes with k arcs, n >= 0, k=0..n*(n-1)/2.

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A097630 Number of unrooted directed trees on n nodes with a red root.

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A380646 Expansion of e.g.f. (1/x) * Series_Reversion( x * exp(-2*x)/(1 + x)^2 ).

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A377360 E.g.f. satisfies A(x) = ( 1 - log(1 - x*A(x)) )^2.

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A097631 Number of unrooted directed trees on n nodes with a green root.

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A152555 Coefficients in a q-analog of the function LambertW(-2x)/(-2x), as a triangle read by rows.