A333361 -id:A333361 - cp's OEIS frontend

A138107 Infinite square array: T(n,k) = number of directed multigraphs with loops with n arcs and k vertices; read by falling antidiagonals.

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 6, 1, 0, 1, 2, 10, 10, 1, 0, 1, 2, 11, 31, 19, 1, 0, 1, 2, 11, 47, 90, 28, 1, 0, 1, 2, 11, 51, 198, 222, 44, 1, 0, 1, 2, 11, 52, 269, 713, 520, 60, 1, 0, 1, 2, 11, 52, 291, 1270, 2423, 1090, 85, 1, 0, 1, 2, 11, 52, 295, 1596, 5776, 7388, 2180, 110, 1, 0

Offset: 0

Benoit Jubin, May 03 2008

Partial sums of the rows of A136564.

			The array begins:
   1, 1,   1,    1,     1,     1,     1,     1,     1, ...
   0, 1,   2,    2,     2,     2,     2,     2,     2, ...
   0, 1,   6,   10,    11,    11,    11,    11,    11, ...
   0, 1,  10,   31,    47,    51,    52,    52,    52, ...
   0, 1,  19,   90,   198,   269,   291,   295,   296,  296, ...
   0, 1,  28,  222,   713,  1270,  1596,  1697,  1719, 1723, ...
   0, 1,  44,  520,  2423,  5776,  8838, 10425, 10922, ...
   0, 1,  60, 1090,  7388, 24032, 46384, ...
   0, 1,  85, 2180, 21003, 93067, ...
   0, 1, 110, 4090, ...
   ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325
R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 [math.CO] (2017) Table 79.

Columns k=0..4 are: A000007, A000012, A005993, A050927, A050929.
Main diagonal is A362387.
Cf. A052171, A136564, A333361.

permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
edges(v,t) = {prod(i=2, #v, prod(j=1, i-1, my(g=gcd(v[i],v[j])); t(v[i]*v[j]/g)^(2*g))) * prod(i=1, #v, t(v[i])^v[i])}
G(n, x)={my(s=0); forpart(p=n, s+=permcount(p)/edges(p,i->1-x^i)); s/n!}
T(n)={Mat(vector(n+1, k, Col(O(y*y^n) + G(k-1, y + O(y*y^n)))))}
{my(A=T(10)); for(n=1, #A, print(A[n,]))} \\ Andrew Howroyd, Oct 22 2019

T(n,k) = Sum_{p=0..k} A136564(n,p).

If k >= 2n, T(n,k) = A052171(n).

More terms from Vladeta Jovovic and Benoit Jubin, Sep 10 2008

A037240 Molien series for 3-D group X1.

Original entry on oeis.org

1, 1, 5, 10, 24, 42, 83, 132, 222, 335, 511, 728, 1047, 1428, 1956, 2586, 3414, 4389, 5638, 7084, 8888, 10966, 13494, 16380, 19841, 23751, 28371, 33566, 39616, 46376, 54177, 62832, 72726, 83661, 96045, 109668, 124999, 141778, 160538, 181006, 203742, 228459, 255788, 285384

Offset: 0

N. J. A. Sloane

nonn

Also multidigraphs with 3 nodes and n arcs. - Vladeta Jovovic, Dec 27 1999

Also preference profiles with 3 alternatives and n agents (IANC model). - Alexander Karpov, Nov 23 2017

Robert Israel, Table of n, a(n) for n = 0..10000
Ö. Egecioglu, Uniform generation of anonymous and neutral preference profiles for social choice rules, Technical Report, UCSB, 2005.
Ö. Egecioglu, Uniform generation of anonymous and neutral preference profiles for social choice rules, Monte Carlo Methods and Applications, 15(3), Jan 2009, 241-255.
Ira Gessel, Combinatorial counting with symmetry, MathOverflow, 2014.
Marko V. Jaric and Joseph L. Birman, Calculation of the Molien generating function for invariants of space groups, J. Math. Phys. 18 (1977), 1459-1465; 2085.
Alexander V. Karpov, An Informational Basis for Voting Rules, NRU Higher School of Economics. Series WP BRP "Economics/EC". 2018. No. 188.
Index entries for Molien series

Column k=3 of A333361.

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( (1 +x^2 +3*x^3 +5*x^4 +x^5 +x^6)/((1-x)*(1-x^2)^3*(1-x^3)^2) )); // G. C. Greubel, Jan 31 2020

S:= series((1+x^2+3*x^3+5*x^4+x^5+ x^6)/(1 - x)/(1 - x^2)^3/(1 - x^3)^2, x, 101):
seq(coeff(S,x,n),n=0..100); # Robert Israel, Nov 22 2017

CoefficientList[Series[(1 +x^2 +3x^3 +5x^4 +x^5 +x^6)/(1-x)/(1-x^2)^3/(1-x^3)^2, {x, 0, 43}], x] (* Michael De Vlieger, Nov 01 2017 *)

Vec((1+x^2+3*x^3+5*x^4+x^5+x^6)/(1-x)/(1-x^2)^3/(1-x^3)^2 + O(x^50)) \\ Michel Marcus, Oct 31 2017

def A037240_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1+x^2+3*x^3+5*x^4+x^5+x^6)/((1-x)*(1-x^2)^3*(1-x^3)^2) ).list()
A037240_list(50) # G. C. Greubel, Jan 31 2020

G.f.: (1 + x^2 + 3*x^3 + 5*x^4 + x^5 + x^6)/((1 - x)*(1 - x^2)^3*(1 - x^3)^2).
From Alexander Karpov, Nov 18 2017: (Start)
if n ==  0  mod 6, a(n) = C(n+5,5)/6 + (n+4)*(n+2)/16 + (n+3)/9;
if n ==  3  mod 6, a(n) = C(n+5,5)/6 + (n+3)/9;
if n == 2,4 mod 6, a(n) = C(n+5,5)/6 + (n+4)*(n+2)/16;
if n == 1,5 mod 6, a(n) = C(n+5,5)/6.
(End)

Terms a(35) and beyond from Alexander Karpov, Oct 29 2017

A052170 Number of directed loopless multigraphs with n arcs.

Original entry on oeis.org

1, 1, 6, 24, 130, 688, 4211, 26840, 184153, 1328155, 10078617, 79926478, 660616432, 5671793248, 50459837996, 464139053799, 4405521306315, 43077862741114, 433275511964227, 4476516495577776, 47451864583578111, 515494036824348917, 5733423512317010811, 65226494052113260251, 758377712833720838677, 9004484581478188581057

Offset: 0

Vladeta Jovovic, Jan 26 2000

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A050535, A007717, A037240, A050930, A333361.

a(n)={polcoef(G(2*n, x + O(x*x^(n))), n)} \\ Needs G from A333361. - Andrew Howroyd, Mar 16 2020

\\ faster, also needs G from A333361.
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
InvEulerMT(u)={my(n=#u, p=log(1+x*Ser(u)), vars=variables(p)); Vec(sum(i=1, n, moebius(i)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i))}
seq(n)={concat([1], EulerT(Vec(-1 + vecsum(InvEulerMT(vector(n+1, k, G(k, y + O(y*y^n))))))))} \\ Andrew Howroyd, Mar 16 2020

a(n) = A333361(n, 2*n). - Andrew Howroyd, Mar 16 2020

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

A050930 Number of directed loopless multigraphs on 4 nodes with n arcs.

Original entry on oeis.org

1, 1, 6, 20, 69, 196, 554, 1368, 3240, 7110, 14896, 29624, 56755, 104468, 186484, 322800, 544590, 896259, 1443834, 2278640, 3531647, 5380064, 8069672, 11926928, 17392983, 25042836, 35637168, 50152068, 69853604, 96344440, 131669538

Offset: 0

Vladeta Jovovic, Dec 30 1999

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Column k=4 of A333361.

Cf. A037240.

G.f.: (2*x^18 + 9*x^17 + 20*x^16 + 46*x^15 + 98*x^14 + 133*x^13 + 196*x^12 + 226*x^11 + 254*x^10 + 240*x^9 + 207*x^8 + 128*x^7 + 92*x^6 + 43*x^5 + 24*x^4 + 8*x^3 + 2*x^2-x + 1)/((x^4-1)^3*(x^3-1)^4*(x^2-1)^3*(x-1)^2).

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A138107 Infinite square array: T(n,k) = number of directed multigraphs with loops with n arcs and k vertices; read by falling antidiagonals.

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A037240 Molien series for 3-D group X1.

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A052170 Number of directed loopless multigraphs with n arcs.

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PARI

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A050930 Number of directed loopless multigraphs on 4 nodes with n arcs.

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