A058391 -id:A058391 - cp's OEIS frontend

A050535 Number of loopless multigraphs on infinite set of nodes with n edges.

1, 1, 3, 8, 23, 66, 212, 686, 2389, 8682, 33160, 132277, 550835, 2384411, 10709827, 49782637, 238998910, 1182772364, 6023860266, 31525780044, 169316000494, 932078457785, 5253664040426, 30290320077851, 178480713438362, 1073918172017297

Offset: 0

Vladeta Jovovic, Dec 29 1999

nonn

Also, a(n) is the number of n-rowed binary matrices with all row sums equal to 2, up to row and column permutation (see Jovovic's formula). Also, a(n) is the limit of A192517(m,n) as m grows. - Max Alekseyev, Oct 18 2017
Row sums of the triangle defined by the Multiset Transformation of A076864,
;
1;
2 1;
5 2 1;
12 8 2 1;
33 22 8 2 1;
103 72 26 8 2 1;
333 229 87 26 8 2 1;
1183 782 295 92 26 8 2 1;
4442 2760 1036 315 92 26 8 2 1;
17576 10270 3735 1129 321 92 26 8 2 1;
72810 39770 13976 4117 1154 321 92 26 8 2 1;
314595 160713 54132 15547 4237 1161 321 92 26 8 2 1;
- R. J. Mathar, Jul 18 2017
Also the number of non-isomorphic set multipartitions (multisets of sets) of {1, 1, 2, 2, 3, 3, ..., n, n}. - Gus Wiseman, Jul 18 2018

			From _Gus Wiseman_, Jul 18 2018: (Start)
Non-isomorphic representatives of the a(3) = 8 set multipartitions of {1, 1, 2, 2, 3, 3}:
  (123)(123)
  (1)(23)(123)
  (12)(13)(23)
  (1)(1)(23)(23)
  (1)(2)(3)(123)
  (1)(2)(13)(23)
  (1)(1)(2)(3)(23)
  (1)(1)(2)(2)(3)(3)
(End)

Frank Harary and Edgar M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 88, Eq. (4.1.18).

Andrew Howroyd, Table of n, a(n) for n = 0..50
George Barnes, Sanjaye Ramgoolam, and Michael Stephanou, Permutation invariant Gaussian matrix models for financial correlation matrices, arXiv:2306.04569 [q-fin.ST], 2023.
Frank Harary, The number of linear, directed, rooted, and connected graphs, Trans. Am. Math. Soc. 78 (1955) 445-463, eq. (24).
Vladeta Jovovic, Number of m-rowed binary matrices with all row sums equal to n, up to row and column permutation
Patrick T. Komiske, Eric M. Metodiev, and Jesse Thaler, Energy flow polynomials: A complete linear basis for jet substructure, arXiv:1712.07124 [hep-ph], 2017.
Tsuyoshi Miezaki, Akihiro Munemasa, Yusaku Nishimura, Tadashi Sakuma, and Shuhei Tsujie, Universal graph series, chromatic functions, and their index theory, arXiv:2403.09985 [math.CO], 2024. See p. 23.

Cf. A001399, A003082, A014395, A014396, A014397, A014398.
Cf. A058389, A050913, A058783, A058390, A058784, A058785, A058391, A058392, A001501, A058528.
Cf. A007716, A007717, A020555, A050535, A076864 (inverse Euler transf.), A076867 (Euler transform) A094574, A316974.

seq[n_] := G[2n, x+O[x]^n, {}] // CoefficientList[#, x]&;
seq[15] (* Jean-François Alcover, Dec 02 2020, using Andrew Howroyd's code for G in A339065 *)

a(n) = A192517(2*n,n) = A192517(m,n) for any m>=2*n. - Max Alekseyev, Oct 18 2017

Euler transform of A076864. - Andrew Howroyd, Oct 23 2019

More terms from Sean A. Irvine, Oct 02 2011

A318951 Array read by rows: T(n,k) is the number of nonisomorphic n X n matrices with nonnegative integer entries and row sums k under row and column permutations, (n >= 1, k >= 0).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 14, 5, 1, 1, 9, 44, 53, 7, 1, 1, 12, 129, 458, 198, 11, 1, 1, 16, 316, 3411, 5929, 782, 15, 1, 1, 20, 714, 19865, 145168, 96073, 3111, 22, 1, 1, 25, 1452, 95214, 2459994, 9283247, 1863594, 12789, 30, 1, 1, 30, 2775, 383714, 30170387, 537001197, 833593500, 42430061, 53836, 42, 1

Offset: 1

Andrew Howroyd, Sep 05 2018

			Array begins:
================================================================
n\k| 0  1    2       3         4            5              6
---|------------------------------------------------------------
1  | 1  1    1       1         1            1              1 ...
2  | 1  2    4       6         9           12             16 ...
3  | 1  3   14      44       129          316            714 ...
4  | 1  5   53     458      3411        19865          95214 ...
5  | 1  7  198    5929    145168      2459994       30170387 ...
6  | 1 11  782   96073   9283247    537001197    19578605324 ...
7  | 1 15 3111 1863594 833593500 189076534322 23361610029905 ...
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Rows 2..6 are A002620(n+2), A058389, A058390, A058391, A058392.

Cf. A304942, A306017.

permcount[v_List] := 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];
K[q_List, t_, k_] := SeriesCoefficient[1/Product[g = GCD[t, q[[j]]]; (1 - x^(q[[j]]/g))^g, {j, 1, Length[q]}], {x, 0, k}];
RowSumMats[n_, m_, k_] := Module[{s = 0}, Do[s += permcount[q]* SeriesCoefficient[Exp[Sum[K[q, t, k]/t*x^t, {t, 1, n}]], {x, 0, n}], {q, IntegerPartitions[m]}]; s/m!];
Table[RowSumMats[n-k, n-k, k], {n, 1, 11}, {k, n-1, 0, -1}] // Flatten (* Jean-François Alcover, Sep 12 2018, after Andrew Howroyd *)

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}
K(q, t, k)={polcoeff(1/prod(j=1, #q, my(g=gcd(t, q[j])); (1 - x^(q[j]/g) + O(x*x^k))^g), k)}
RowSumMats(n, m, k)={my(s=0); forpart(q=m, s+=permcount(q)*polcoeff(exp(sum(t=1, n, K(q, t, k)/t*x^t) + O(x*x^n)), n)); s/m!}
for(n=1, 8, for(k=0, 6, print1(RowSumMats(n, n, k), ", ")); print)

A058389 Number of 3 X 3 matrices with nonnegative integer entries and all row sums equal to n, up to row and column permutation.

Original entry on oeis.org

1, 3, 14, 44, 129, 316, 714, 1452, 2775, 4963, 8478, 13838, 21827, 33306, 49504, 71754, 101871, 141807, 194128, 261570, 347633, 456026, 591384, 758596, 963657, 1212861, 1513806, 1874440, 2304225, 2813030, 3412466, 4114608, 4933519

Offset: 0

Vladeta Jovovic, Nov 24 2000

Andrew Howroyd, Table of n, a(n) for n = 0..1000
V. Jovovic, Illustration of initial terms
V. Jovovic, Number of m x m nonnegative integer matrices with all row sums equal to n, up to row and column permutation.

Row 3 of A318951.
Cf. A002817, A052282, A058390, A058391, A058392.
Cf. A001501, A050535, A050913, A058528, A058783, A058784, A058785.

a[n_] := (m = Mod[n, 6]; (n^3 + 9*n^2 + 39*n + 120)*n^3 + Which[m == 0, 12*(23*n^2 + 32*n + 24), m == 1 || m == 5, 249*n^2 + 303*n + 143, m == 2 || m == 4, 4*(69*n^2 + 96*n + 56), m == 3, 3*(83*n^2 + 101*n + 69)])/288; Table[a[n], {n, 0, 32}] (* Jean-François Alcover, Oct 12 2011, after Vladeta Jovovic *)

\\ See A318951 for RowSumMats
a(n)=RowSumMats(3, 3, n); \\ Andrew Howroyd, Sep 05 2018

a(n) = (1/6)*(C(C(n + 2, 2) + 2, 3) + 3/2*floor((n + 2)/2)*(C(n + 2, 2) - floor((n + 2)/2)) + 3*C(floor((n + 2)/2) + 2, 3) + 2*floor(C(n + 2, 2)/3) + 2*C(C(n + 2, 2) - 3*floor(C(n + 2, 2)/3) + 2, 3)).

Empirical G.f.: -(x^8 + 3*x^7 + 14*x^6 + 12*x^5 + 15*x^4 + 9*x^3 + 5*x^2 + 1) / ((x-1)^7*(x+1)^3*(x^2+x+1)). - Colin Barker, Dec 27 2012

More terms from Marc LeBrun, Dec 11 2000

A058390 Number of 4 X 4 matrices with nonnegative integer entries and all row sums equal to n, up to row and column permutation.

Original entry on oeis.org

1, 5, 53, 458, 3411, 19865, 95214, 383714, 1346183, 4202086, 11905966, 31061806, 75533056, 172800689, 374861365, 775978710, 1541027694, 2949003213, 5458806804, 9805626744, 17140511056

Offset: 0

Vladeta Jovovic, Nov 24 2000

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Vladeta Jovovic, Number of m x m nonnegative integer matrices with all row sums equal to n, up to row and column permutation.

Row 4 of A318951.

Cf. A002620, A058389, A058391, A058392, A001496, A052280, A052281.

(* See A318951 for RowSumMats *)
a[n_] := RowSumMats[4, 4, n];
a /@ Range[0, 20] (* Jean-François Alcover, Sep 15 2019, after Andrew Howroyd *)

\\ See A318951 for RowSumMats
a(n)=RowSumMats(4, 4, n); \\ Andrew Howroyd, Sep 05 2018

A058392 Number of 6 X 6 matrices with nonnegative integer entries and all row sums equal to n, up to row and column permutation.

Original entry on oeis.org

1, 11, 782, 96073, 9283247, 537001197, 19578605324, 487615778173, 8892272235593, 125319645293555, 1423054983691408, 13451239365449764, 108603794657349271, 764673059329865921, 4775254548845993462, 26820549989969591853, 137072193873357150230, 643738505766475169048

Offset: 0

Vladeta Jovovic, Nov 24 2000

nonn

Andrew Howroyd, Table of n, a(n) for n = 0..1000
V. Jovovic, Number of m x m nonnegative integer matrices with all row sums equal to n, up to row and column permutation.

Row 6 of A318951.

Cf. A002620, A058389-A058391, A003439, A005467.

\\ See A318951 for RowSumMats
a(n)=RowSumMats(6, 6, n); \\ Andrew Howroyd, Sep 05 2018

Terms a(15) and beyond from Andrew Howroyd, Sep 05 2018

A003438 Number of 5 X 5 matrices with nonnegative integer entries and row and column sums equal to n.

Original entry on oeis.org

1, 120, 6210, 153040, 2224955, 22069251, 164176640, 976395820, 4855258305, 20856798285, 79315936751, 272095118010, 854560160105, 2486299719645, 6765755480415, 17356306529251, 42250330784180, 98137852369965

Offset: 0

N. J. A. Sloane

Number of 5 X 5 stochastic matrices of integers.

D. M. Jackson and G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4 (1975), 474-477.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Wadsworth, Vol. 1, 1986, p. 234.

T. D. Noe, Table of n, a(n) for n = 0..1000
D. M. Jackson & G. H. J. van Rees, The enumeration of generalized double stochastic nonnegative integer square matrices, SIAM J. Comput., 4.4 (1975), 474-477. (Annotated scanned copy)
M. L. Stein and P. R. Stein, Enumeration of Stochastic Matrices with Integer Elements, Report LA-4434, Los Alamos Scientific Laboratory of the University of California, Los Alamos, NM, Jun 1970. [Annotated scanned copy]
Index entries for linear recurrences with constant coefficients, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).

Cf. A002817, A001496, A019298.
Cf. A001496, A005466, A058391.
Row n=5 of A257493.

CoefficientList[Series[(1+103x+4306x^2+63110x^3+388615x^4+1115068x^5+ 1575669x^6+1115068x^7+388615x^8+63110x^9+4306x^10+103x^11+x^12)/ (1-x)^17,{x,0,30}],x] (* Harvey P. Dale, Aug 17 2013 *)

G.f.: (1 + 103*x + 4306*x^2 + 63110*x^3 + 388615*x^4 + 1115068*x^5 + 1575669*x^6 + 1115068*x^7 + 388615*x^8 + 63110*x^9 + 4306*x^10 + 103*x^11 + x^12)/(1-x)^17.

a(n) = Sum_{j=0..6} A005466(j) * binomial(4+j+n, 4+2*j). - Andrew Howroyd, Apr 09 2020

More terms from Vladeta Jovovic, Feb 06 2000

A110058 Number of nonnegative integer matrices of order n for which all row and column sums equal n.

Original entry on oeis.org

1, 1, 3, 55, 10147, 22069251, 602351808741, 215717608046511873, 1046591482728407939338275, 70417932475495769964322670258947, 66880713903767740581650957184096513655153, 909176713758393122455793478657031533216492953328933, 178876969166665269546249744608783223036842010760723370462856181, 514016665650183402309555825250370336139392333285719205357202846243695510965

Offset: 0

Brendan McKay, Sep 04 2005

nonn

Computed by a method that involves summing a multivariate generating function over roots of unity.

			a(2) = 3 due to the matrices [1,1 | 1,1], [0,2 | 2,0] and [2,0 | 0,2].

E. R. Canfield, B. D. McKay, Asymptotic enumeration of integer matrices with large equal row and column sums, Combinatorica 30 (6) (2010) 655-680

Cf. A058407, A058410, A058391.

Main diagonal of A257493 and A333901.

from sage.combinat.integer_matrices import IntegerMatrices
[IntegerMatrices([n]*n, [n]*n).cardinality() for n in (0..6)] # Freddy Barrera, Dec 27 2018

log a(n) = 2(log 2)*n^2 - n*(log n) - n*(log 4*Pi) + (log n) + O(1). - Igor Pak, May 15 2019

a(0)=1 prepended by Alois P. Heinz, Apr 26 2015

cp's OEIS Frontend

A050535 Number of loopless multigraphs on infinite set of nodes with n edges.

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A318951 Array read by rows: T(n,k) is the number of nonisomorphic n X n matrices with nonnegative integer entries and row sums k under row and column permutations, (n >= 1, k >= 0).

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A058389 Number of 3 X 3 matrices with nonnegative integer entries and all row sums equal to n, up to row and column permutation.

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A058390 Number of 4 X 4 matrices with nonnegative integer entries and all row sums equal to n, up to row and column permutation.

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A058392 Number of 6 X 6 matrices with nonnegative integer entries and all row sums equal to n, up to row and column permutation.

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A003438 Number of 5 X 5 matrices with nonnegative integer entries and row and column sums equal to n.

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A110058 Number of nonnegative integer matrices of order n for which all row and column sums equal n.

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Sage

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