A358451 -id:A358451 - cp's OEIS frontend

A003085 Number of weakly connected digraphs with n unlabeled nodes.

1, 2, 13, 199, 9364, 1530843, 880471142, 1792473955306, 13026161682466252, 341247400399400765678, 32522568098548115377595264, 11366712907233351006127136886487, 14669074325902449468573755897547924182

Offset: 1

N. J. A. Sloane

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, pp. 124 and 241.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Keith Briggs, Table of n, a(n) for n = 1..64
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Alexander G. Ginsberg, Firing-Rate Models in Computational Neuroscience: New Applications and Methodologies, Ph. D. Dissertation, Univ. Michigan, 2023. See p. 7.
Martin Golubitsky and Yangyang Wang, Infinitesimal homeostasis in three-node input-output networks, Journal of Mathematical Biology (2020) Vol. 80, 1163-1185.
A. Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.
X. Li, D. S. Stones, H. Wang, H. Deng, X. Liu and G. Wang, NetMODE: Network Motif Detection without Nauty, PLoS ONE 7(12): e50093. doi:10.1371/journal.pone.0050093. - From _N. J. A. Sloane_, Feb 02 2013
Eric Weisstein's World of Mathematics, Weakly Connected Digraph

Row sums of A054733.
Column sums of A350789.
The labeled case is A003027.
Cf. A000273, A003084, A035512 (strongly connected).

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(A000273):
seq(a(n), n = 1..13); # Peter Luschny, Nov 21 2022

Needs["Combinatorica`"]; d[n_] := GraphPolynomial[n, x, Directed] /. x -> 1; max = 13; se = Series[ Sum[a[n]*x^n/n, {n, 1, max}] - Log[1 + Sum[ d[n]*x^n, {n, 1, max}]], {x, 0, max}]; sol = SolveAlways[ se == 0, x]; Do[ A003084[n] = a[n] /. sol[[1]], {n, 1, max}]; ClearAll[a, d]; a[n_] := (1/n)*Sum[ MoebiusMu[ n/d ] * A003084[d], {d, Divisors[n]} ]; Table[ a[n], {n, 1, max}] (* Jean-François Alcover, Feb 01 2012, after formula *)
terms = 13;
permcount[v_] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
edges[v_] := Sum[2*GCD[v[[i]], v[[j]]], {i, 2, Length[v]}, {j, 1, i - 1}] + Total[v - 1];
d[n_] := (s = 0; Do[s += permcount[p]*2^edges[p], {p, IntegerPartitions[n]} ]; s/n!);
A003084 = CoefficientList[Log[Sum[d[n] x^n, {n, 0, terms+1}]] + O[x]^(terms + 1), x] Range[0, terms] // Rest;
a[n_] := (1/n)*Sum[MoebiusMu[n/d] * A003084[[d]], {d, Divisors[n]}];
Table[a[n], {n, 1, terms}] (* Jean-François Alcover, Aug 30 2019, after Andrew Howroyd in A003084 *)

from functools import lru_cache
from itertools import product, combinations
from fractions import Fraction
from math import prod, gcd, factorial
from sympy import mobius, divisors
from sympy.utilities.iterables import partitions
def A003085(n):
    @lru_cache(maxsize=None)
    def b(n): return int(sum(Fraction(1<Chai Wah Wu, Jul 05 2024

a(n) = (1/n)*Sum_{d|n} mu(n/d)*A003084(d), where mu is Moebius function.

Inverse Euler transform of A000273. - Andrew Howroyd, Dec 27 2021

More terms from Vladeta Jovovic, Jan 09 2000

A320779 Inverse Euler transform of the number of divisors function A000005.

Original entry on oeis.org

1, 1, 0, 0, -1, 1, -1, 0, 1, -1, 0, 1, -1, -1, 2, 1, -2, -2, 2, 3, -4, 0, 3, -3, 3, -2, -2, 2, 1, 7, -15, 0, 17, -11, -1, 0, 9, -4, -18, 26, -10, -10, 24, -17, -15, 21, 27, -42, -37, 69, 43, -113, -11, 149, -98, -24, 67, -57, 24, -53, 213, -243, -193, 704

Offset: 1

Gus Wiseman, Oct 22 2018

sign

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n).

Alois P. Heinz, Table of n, a(n) for n = 1..5000
OEIS Wiki, Euler transform

Cf. A000005.

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> ifelse(n=0, 1, NumberTheory:-SumOfDivisors(n, 0))):
seq(a(n), n = 1..64); # Peter Luschny, Nov 21 2022

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
EulerInvTransform[Table[DivisorSigma[0,n],{n,100}]]

from functools import lru_cache
from sympy import mobius, divisors, divisor_count
def A320779(n):
    @lru_cache(maxsize=None)
    def b(n): return divisor_count(n)
    @lru_cache(maxsize=None)
    def c(n): return n*b(n)-sum(c(k)*b(n-k) for k in range(1,n))
    return sum(mobius(d)*c(n//d) for d in divisors(n,generator=True))//n # Chai Wah Wu, Jul 15 2024

A320781 Inverse Euler transform of the Moebius function A008683.

Original entry on oeis.org

1, -2, 0, 0, -1, 2, -4, 5, -7, 9, -10, 7, -5, -2, 19, -44, 70, -103, 138, -166, 154, -83, -70, 346, -797, 1413, -2160, 2931, -3479, 3380, -2080, -1259, 7593, -17743, 32014, -49818, 68683, -82985, 82807, -53462, -24942, 176139, -422887, 777357, -1226688

Offset: 1

Gus Wiseman, Oct 22 2018

sign

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n).

OEIS Wiki, Euler transform

Cf. A008683,

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> ifelse(n=0, 1, NumberTheory:-Moebius(n))):
seq(a(n), n = 1..45); # Peter Luschny, Nov 21 2022

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
EulerInvTransform[Table[MoebiusMu[n],{n,30}]]

from functools import lru_cache
from sympy import mobius, divisors
def A320781(n):
    @lru_cache(maxsize=None)
    def b(n): return mobius(n)
    @lru_cache(maxsize=None)
    def c(n): return n*b(n)-sum(c(k)*b(n-k) for k in range(1,n))
    return sum(b(d)*c(n//d) for d in divisors(n,generator=True))//n # Chai Wah Wu, Jul 15 2024

A320780 Inverse Euler transform of the sum-of-divisors or sigma function A000203.

Original entry on oeis.org

1, 2, 1, 0, -3, 1, -1, 1, 3, -5, -1, 4, 3, -3, -7, 8, 1, -9, 7, 8, -13, -12, 27, 7, -19, -14, 11, -17, -25, 198, -81, -312, 89, 326, 325, -739, -275, 572, -255, 1287, -453, -2062, -583, 2155, 5985, -6725, -6661, 6968, 3045, 3876, -7205, -2773, -5447, -4902

Offset: 1

Gus Wiseman, Oct 22 2018

sign

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n).

OEIS Wiki, Euler transform

Cf. A000203.

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> ifelse(n=0, 1, NumberTheory:-SumOfDivisors(n, 1))):
seq(a(n), n = 1..54); # Peter Luschny, Nov 21 2022

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
EulerInvTransform[Table[DivisorSigma[1,n],{n,30}]]

A320778 Inverse Euler transform of the Euler totient function phi = A000010.

Original entry on oeis.org

1, 1, 0, 1, 0, 2, -3, 4, -4, 4, -9, 14, -19, 30, -42, 50, -76, 128, -194, 286, -412, 598, -909, 1386, -2100, 3178, -4763, 7122, -10758, 16414, -25061, 38056, -57643, 87568, -133436, 203618, -311128, 475536, -726355, 1109718, -1697766, 2601166, -3987903, 6114666

Offset: 0

Gus Wiseman, Oct 22 2018

sign

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.

Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.
Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.
Inverse Euler transforms: A059966, A320767, A320776, A320777, A320779, A320780, A320781, A320782.

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> ifelse(n=0, 1, NumberTheory:-Totient(n))):
seq(a(n), n = 0..43); # Peter Luschny, Nov 21 2022

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
EulerInvTransform[Array[EulerPhi,30]]

A085686 Inverse Euler transform of Bell numbers.

Original entry on oeis.org

1, 1, 3, 9, 34, 135, 610, 2965, 15612, 87871, 526274, 3334850, 22270254, 156172689, 1146640394, 8791424549, 70227355786, 583283741066, 5027823752930, 44903579626132, 414877600876638, 3959945232723603, 38996757506464858, 395749369598406027, 4134132167178705732

Offset: 1

N. J. A. Sloane, Jul 18 2003

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..300
Nicolas Andrews, Lucas Gagnon, Félix Gélinas, Eric Schlums, and Mike Zabrocki, When are Hopf algebras determined by integer sequences?, arXiv:2505.06941 [math.CO], 2025. See p. 3.

Cf. A000110, A305846.

read transforms; A := series(exp(exp(x)-1),x,60); A000110 := n->n!*coeff(A,x,n); [seq(A000110(i),i=1..30)]; EULERi(%);
# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(combinat:-bell):
seq(a(n), n = 1..25); # Peter Luschny, Nov 21 2022

n=24; eq[0] = Rest[ Thread[ CoefficientList[ 1 + Series[ Sum[ BellB[k]*x^k, {k, 1, n}] - Product[1/(1-x^k)^a[k], {k, 1, n}], {x, 0, n}], x] == 0]]; s[1] = First[ Solve[ First[eq[0]], a[1]]]; Do[eq[k] = Rest[eq[k-1]] /. s[k]; s[k+1] = First[ Solve[ First[eq[k]], a[k+1]]], {k, 1, n-1}]; Table[a[k], {k, 1, n}] /. Flatten[Table[s[k], {k, 1, n}]] (* Jean-François Alcover, Jul 26 2011 *)
bb = Array[BellB, n = 25]; s = {}; For[i = 1, i <= n, i++, AppendTo[s, i* bb[[i]] - Sum[s[[d]]*bb[[i-d]], {d, i-1}]]]; Table[Sum[If[Divisible[i, d], MoebiusMu[i/d], 0]*s[[d]], {d, 1, i}]/i, {i, n}] (* Jean-François Alcover, Apr 15 2016 *)

A112354 Inverse Euler transform of n!. Also the number of sequences of permutations with no global descents which are Lyndon (smallest in lexicographic order of all cyclic shifts of the sequences) where the size of the sequence = sum of sizes of the permutations.

Original entry on oeis.org

1, 1, 4, 17, 92, 572, 4156, 34159, 314368, 3199844, 35703996, 433421495, 5687955724, 80256874912, 1211781887796, 19496946534720, 333041104402860, 6019770246910128, 114794574818830716, 2303332661416242633, 48509766592884311132, 1069983257387132347080

Offset: 1

Mike Zabrocki, Sep 05 2005

nonn

			a(3) = 4 because (123), (213), (132) and (1,21) are all Lyndon.
a(4) = 17 because there are 13 permutations with no global descents of size 4 and (1,123), (1,213), (1,132) are all Lyndon.
a(5) = 92 = 71 permutations with no global descents+13 sequences of the form (1,pi) where pi in S_4 with no global descents+(1,1,1,21),(1,21,21),(1,1,123),(1,1,213),(1,1,132),(21,123),(21,213),(21,132).

Seiichi Manyama, Table of n, a(n) for n = 1..449 (terms 1..200 from Alois P. Heinz)
M. Aguiar and A. Lauve, Antipode and Convolution Powers of the Identity in Graded Connected Hopf Algebras, FPSAC 2013 Paris, France DMTCS Proc. AS, 2013, 1083-1094.
M. Aguiar, A. Lauve, The characteristic polynomial of the Adams operators on graded connected Hopf algebras, arXiv:1403.7584 [math.RA], 2014.

Cf. A003319, A000142, A305786.

read transforms; EULERi([seq(n!,n=1..30)]);
# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(factorial):
seq(a(n), n = 1..22); # Peter Luschny, Nov 21 2022

ff = Range[n = 22]!; s = {}; For[i = 1, i <= n, i++, AppendTo[s, i*ff[[i]] - Sum[s[[d]]*ff[[i-d]], {d, i-1}]]]; Table[Sum[If[Divisible[i, d], MoebiusMu[i/d], 0]*s[[d]], {d, 1, i}]/i, {i, n}] (* Jean-François Alcover, Apr 15 2016 *)

Product_{k>=1} 1/(1-x^k)^{a(k)} = Sum_{n>=0} n! x^n.

a(n) ~ n! * (1 - 1/n - 1/n^2 - 4/n^3 - 23/n^4 - 171/n^5 - 1542/n^6 - 16241/n^7 - 194973/n^8 - 2622610/n^9 - 39027573/n^10 - ...), for coefficients see A113869. - Vaclav Kotesovec, Sep 04 2014, extended Nov 27 2020

A320776 Inverse Euler transform of the number of prime factors (with multiplicity) function A001222.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, -1, -1, 0, 1, 0, -1, -1, -1, 1, 3, 3, -2, -5, -4, 0, 7, 7, 0, -9, -10, 2, 15, 15, -3, -27, -30, 3, 46, 51, 1, -71, -91, -7, 117, 157, 23, -194, -265, -57, 318, 465, 111, -536, -821, -230, 893, 1456, 505, -1485, -2559, -1036, 2433, 4483, 2022

Offset: 0

Gus Wiseman, Oct 22 2018

sign

The Euler transform of a sequence q is the sequence of coefficients of x^n, n > 0, in the expansion of Product_{n > 0} 1/(1 - x^n)^q(n). The constant term 1 is sometimes taken to be the zeroth part of the Euler transform.

Number theoretical functions: A000005, A000010, A000203, A001055, A001221, A001222, A008683, A010054.
Euler transforms: A000081, A001970, A006171, A007294, A061255, A061256, A061257, A073576, A117209, A293548, A293549.
Inverse Euler transforms: A059966, A320767, A320777, A320778, A320779, A320780, A320781, A320782.

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(n -> ifelse(n=0, 1, NumberTheory:-NumberOfPrimeFactors(n))):
seq(a(n), n = 0..59); # Peter Luschny, Nov 21 2022

EulerInvTransform[{}]={};EulerInvTransform[seq_]:=Module[{final={}},For[i=1,i<=Length[seq],i++,AppendTo[final,i*seq[[i]]-Sum[final[[d]]*seq[[i-d]],{d,i-1}]]];
Table[Sum[MoebiusMu[i/d]*final[[d]],{d,Divisors[i]}]/i,{i,Length[seq]}]];
EulerInvTransform[Array[PrimeOmega,100]]

A095993 Inverse Euler transform of the ordered Bell numbers A000670.

Original entry on oeis.org

1, 1, 2, 10, 59, 446, 3965, 41098, 484090, 6390488, 93419519, 1498268466, 26159936547, 494036061550, 10035451706821, 218207845446062, 5057251219268460, 124462048466812950, 3241773988588098756, 89093816361187396674, 2576652694087142999421

Offset: 0

Mike Zabrocki, Jul 18 2004

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..423

Cf. A000670, A085686, A095989.

read transforms; A000670 := proc(n) option remember; local k; if n <=1 then 1 else add(binomial(n,k)*A000670(n-k),k=1..n); fi; end; [seq(A000670(i),i=1..30)]; EULERi(%);
# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(A000670):
seq(a(n), n = 0..22); # Peter Luschny, Nov 21 2022

max = 25; b[0] = 1; b[n_] := b[n] = Sum[Binomial[n, k]*b[n-k], {k, 1, n}]; bb = Array[b, max]; s = {}; For[i=1, i <= max, i++, AppendTo[s, i*bb[[i]] - Sum[s[[d]]*bb[[i-d]], {d, i-1}]]]; a[0] = 1; a[n_] := Sum[If[Divisible[ n, d], MoebiusMu[n/d], 0]*s[[d]], {d, 1, n}]/n; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Feb 25 2017 *)

Product(1/(1-q^n)^(a(n)), n >=1) = sum(A000670(k)*q^k, k>=0).

a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Oct 09 2019

a(0)=1 inserted by Alois P. Heinz, Feb 20 2017

A095975 -a(n) is inverse EULER transform of -A000041(n).

Original entry on oeis.org

1, 2, 5, 11, 27, 60, 147, 344, 839, 2031, 5017, 12379, 30921, 77407, 195121, 493451, 1253613, 3194303, 8166757, 20933754, 53798919, 138566312, 357647565, 924834079, 2395702801, 6215748612, 16150985071, 42024182520, 109485000777, 285578913962, 745728542725

Offset: 1

Vladeta Jovovic, Jul 20 2004

Alois P. Heinz, Table of n, a(n) for n = 1..1000
N. J. A. Sloane, Transforms

Cf. A055887, A246828.

with(numtheory): b:= proc(n) option remember; `if`(n=0,1, add(add(d, d=divisors(j)) *b(n-j), j=1..n)/n) end: c:= proc(n) option remember; local j; add(c(j) *b(n-j), j=1..n-1)-n*b(n) end: a:= -proc(n) option remember; local d; `if`(n=0,1, add(mobius(n/d)*c(d), d=divisors(n))/n) end: seq(a(n), n=1..40); # Alois P. Heinz, Sep 09 2008
# The function EulerInvTransform is defined in A358451.
a := -EulerInvTransform(n -> -combinat:-numbpart(n)):
seq(a(n), n = 1..31); # Peter Luschny, Nov 21 2022

b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d, {d, Divisors[j]}]*b[n-j], {j, 1, n}]/n]; c[n_] := c[n] = Sum[c[j]*b[n-j], {j, 1, n-1}] - n*b[n]; a[n_] := -If[n == 0, 1, Sum[MoebiusMu[n/d]*c[d], {d, Divisors[n]}]/n]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Feb 24 2015, after Alois P. Heinz *)

Moebius transform of A055890(n).

a(n) ~ d^n / n, d = 2.69832910647421123126399866618837... (see A246828). - Vaclav Kotesovec, Aug 25 2014

More terms from Alois P. Heinz, Sep 09 2008

cp's OEIS Frontend

A003085 Number of weakly connected digraphs with n unlabeled nodes.

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A320779 Inverse Euler transform of the number of divisors function A000005.

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A320781 Inverse Euler transform of the Moebius function A008683.

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A320780 Inverse Euler transform of the sum-of-divisors or sigma function A000203.

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A320778 Inverse Euler transform of the Euler totient function phi = A000010.

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A085686 Inverse Euler transform of Bell numbers.

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A112354 Inverse Euler transform of n!. Also the number of sequences of permutations with no global descents which are Lyndon (smallest in lexicographic order of all cyclic shifts of the sequences) where the size of the sequence = sum of sizes of the permutations.

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A320776 Inverse Euler transform of the number of prime factors (with multiplicity) function A001222.

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