A323654 -id:A323654 - cp's OEIS frontend

A052847 G.f.: 1 / Product_{k>=1} (1-x^k)^(k-1).

1, 0, 1, 2, 4, 6, 12, 18, 33, 52, 88, 138, 229, 354, 568, 880, 1378, 2110, 3260, 4942, 7527, 11320, 17031, 25394, 37842, 55956, 82630, 121300, 177677, 258980, 376626, 545352, 787784, 1133764, 1627657, 2329020, 3324559, 4731396, 6717774, 9512060

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Euler transform of sequence [0,1,2,3,...]. - Michael Somos, Jul 02 2004
Number of partitions of n objects of 2 colors, where each part must contain at least one of each color. - Franklin T. Adams-Watters, Jan 23 2006
Number of partitions of n without 1s, one kind of 2s, two kinds of 3s, etc. - Joerg Arndt, Jul 31 2011
From Vaclav Kotesovec, Oct 17 2015: (Start)
In general, if v>=0 and g.f. = Product_{k>=1} 1/(1-x^(k+v))^k, then a(n) ~ d1(v) * n^(v^2/6 - 25/36) * exp(-Pi^4 * v^2 / (432*Zeta(3)) + 3*Zeta(3)^(1/3) * n^(2/3)/2^(2/3) - v * Pi^2 * n^(1/3) / (3 * 2^(4/3) * Zeta(3)^(1/3))) / (sqrt(3*Pi) * 2^(v^2/6 + 11/36) * Zeta(3)^(v^2/6 - 7/36)), where Zeta(3) = A002117.
d1(v) = exp(Integral_{x=0..infinity} (1/(x*exp((v-1)*x) * (exp(x)-1)^2) - (6*v^2-1) / (12*x*exp(x)) + v/x^2 - 1/x^3) dx).
d1(v) = (exp(Zeta'(-1) - v*Zeta'(0))) / Product_{j=0..v-1} j!, where Zeta'(0) = -A075700, Zeta'(-1) = A084448 and Product_{j=0..v-1} j! = A000178(v-1).
d1(v) = exp(1/12) * (2*Pi)^(v/2) / (A * G(v+1)), where A = A074962 is the Glaisher-Kinkelin constant and G is the Barnes G-function.
(End)

			1 + x^2 + 2*x^3 + 4*x^4 + 6*x^5 + 12*x^6 + 18*x^7 + 33*x^8 + 52*x^9 + ...
From _Gus Wiseman_, Jan 22 2019: (Start)
The partitions described in Franklin T. Adams-Watters's comment are (n = 2 through 6):
  {{12}}  {{112}}  {{1112}}    {{11112}}    {{111112}}
          {{122}}  {{1122}}    {{11122}}    {{111122}}
                   {{1222}}    {{11222}}    {{111222}}
                   {{12}{12}}  {{12222}}    {{112222}}
                               {{12}{112}}  {{122222}}
                               {{12}{122}}  {{112}{112}}
                                            {{112}{122}}
                                            {{12}{1112}}
                                            {{12}{1122}}
                                            {{12}{1222}}
                                            {{122}{122}}
                                            {{12}{12}{12}}
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 815
Vaclav Kotesovec, Graph - The asymptotic ratio

Cf. A005380, A000203, A001157, A052812.
Cf. A000219 (v=0), A052847 (v=1), A263358 (v=2), A263359 (v=3), A263360 (v=4), A263361 (v=5), A263362 (v=6), A263363 (v=7), A263364 (v=8).
Cf. A054974, A321407, A321760, A323654, A323655, A323656.

spec := [S,{B=Sequence(Z,1 <= card),C=Prod(B,B),S= Set(C)},unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);
with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:=etr(n-> n-1): seq(a(n), n=0..50); # Vaclav Kotesovec, Mar 04 2015 after Alois P. Heinz

Clear[a]; a[n_]:= a[n] = 1/n*Sum[(DivisorSigma[2,k]-DivisorSigma[1,k])*a[n-k],{k,1,n}]; a[0]=1; Table[a[n],{n,0,100}] (* Vaclav Kotesovec, Mar 04 2015 *)
nmax = 40; CoefficientList[Series[Product[1/(1-x^(k+1))^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 16 2015 *)

{a(n) = if( n<0, 0, polcoeff( 1 / prod(k=1, n, (1 - x^k + x*O(x^n))^(k-1)), n))}

a(n) = 1/n*Sum_{k=1..n} (sigma[2](k)-sigma[1](k))*a(n-k).
G.f.: exp( Sum_{k>0} ( x^k / (1 - x^k) )^2 / k ).
G.f.: exp( sum(n>=0, (sigma[2](n)-sigma[1](n)) *x^n/n ) ). - Joerg Arndt, Jul 31 2011
a(n) ~ 2^(1/36) * Zeta(3)^(1/36) * exp(1/12 - Pi^4/(432*Zeta(3)) - Pi^2 * n^(1/3) / (3 * 2^(4/3) * Zeta(3)^(1/3)) + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A * 3^(1/2) * n^(19/36)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Mar 07 2015

Edited by Vladeta Jovovic, Sep 10 2002

A054974 Number of nonnegative integer 2 X 2 matrices with no zero rows or columns and with sum of elements equal to n, up to row and column permutation.

Original entry on oeis.org

1, 2, 6, 9, 17, 23, 36, 46, 65, 80, 106, 127, 161, 189, 232, 268, 321, 366, 430, 485, 561, 627, 716, 794, 897, 988, 1106, 1211, 1345, 1465, 1616, 1752, 1921, 2074, 2262, 2433, 2641, 2831, 3060, 3270, 3521, 3752, 4026, 4279, 4577, 4853, 5176, 5476, 5825, 6150

Offset: 2

Vladeta Jovovic, May 28 2000

From Gus Wiseman, Jan 22 2019: (Start)
Also the number of non-isomorphic multiset partitions of weight n with exactly 2 distinct vertices and exactly 2 (not necessarily distinct) edges. For example, non-isomorphic representatives of the a(2) = 1 through a(5) = 9 multiset partitions are:
  {{1}{2}}  {{1}{22}}  {{1}{122}}  {{11}{122}}
            {{2}{12}}  {{11}{22}}  {{1}{1222}}
                       {{12}{12}}  {{11}{222}}
                       {{1}{222}}  {{12}{122}}
                       {{12}{22}}  {{1}{2222}}
                       {{2}{122}}  {{12}{222}}
                                   {{2}{1122}}
                                   {{2}{1222}}
                                   {{22}{122}}
(End)

			There are 9 nonnegative integer 2 X 2 matrices with no zero rows or columns and with sum of elements equal to 5, up to row and column permutation:
[0 1] [0 1] [0 1] [0 1] [0 2] [0 2] [0 2] [0 3] [1 1]
[1 3] [2 2] [3 1] [4 0] [1 2] [2 1] [3 0] [1 1] [1 2].

Colin Barker, Table of n, a(n) for n = 2..1000
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Column k=2 of A321615.

Cf. A007716, A052847, A053307, A323654, A323655, A323656.

gf := -x^2*(x^3-x^2-1)/((x^2-1)^2*(x-1)^2): s := series(gf, x, 101): for i from 2 to 100 do printf(`%d,`,coeff(s,x,i)) od:

Vec(-x^2*(x^3-x^2-1) / ((x^2-1)^2*(x-1)^2) + O(x^60)) \\ Colin Barker, Jan 16 2017

G.f.: -x^2*(x^3-x^2-1) / ((x^2-1)^2*(x-1)^2).
From Colin Barker, Jan 16 2017: (Start)
a(n) = (6 - 6*(-1)^n + (9*(-1)^n-17)*n + 12*n^2 + 2*n^3) / 48.
a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>7.
(End)

More terms from James Sellers, May 29 2000

A323655 Number of non-isomorphic multiset partitions of weight n with at most 2 distinct vertices, or with at most 2 (not necessarily distinct) edges.

Original entry on oeis.org

1, 1, 4, 7, 19, 35, 80, 149, 307, 566, 1092, 1974, 3643, 6447, 11498, 19947, 34636, 58974, 100182, 167713, 279659, 461056, 756562, 1230104, 1990255, 3195471, 5105540, 8103722, 12801925, 20107448, 31439978, 48907179, 75755094, 116797754, 179354540, 274253042

Offset: 0

Gus Wiseman, Jan 22 2019

nonn

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Also the number of nonnegative integer matrices with only one or two columns, no zero rows or columns, and sum of entries equal to n, up to row and column permutations.

			Non-isomorphic representatives of the a(1) = 1 through a(4) = 19 multiset partitions with at most 2 distinct vertices:
  {{1}}  {{11}}    {{111}}      {{1111}}
         {{12}}    {{122}}      {{1122}}
         {{1}{1}}  {{1}{11}}    {{1222}}
         {{1}{2}}  {{1}{22}}    {{1}{111}}
                   {{2}{12}}    {{11}{11}}
                   {{1}{1}{1}}  {{1}{122}}
                   {{1}{2}{2}}  {{11}{22}}
                                {{12}{12}}
                                {{1}{222}}
                                {{12}{22}}
                                {{2}{122}}
                                {{1}{1}{11}}
                                {{1}{1}{22}}
                                {{1}{2}{12}}
                                {{1}{2}{22}}
                                {{2}{2}{12}}
                                {{1}{1}{1}{1}}
                                {{1}{1}{2}{2}}
                                {{1}{2}{2}{2}}
Non-isomorphic representatives of the a(1) = 1 through a(4) = 19 multiset partitions with at most 2 edges:
  {{1}}  {{11}}    {{111}}    {{1111}}
         {{12}}    {{122}}    {{1122}}
         {{1}{1}}  {{123}}    {{1222}}
         {{1}{2}}  {{1}{11}}  {{1233}}
                   {{1}{22}}  {{1234}}
                   {{1}{23}}  {{1}{111}}
                   {{2}{12}}  {{11}{11}}
                              {{1}{122}}
                              {{11}{22}}
                              {{12}{12}}
                              {{1}{222}}
                              {{12}{22}}
                              {{1}{233}}
                              {{12}{33}}
                              {{1}{234}}
                              {{12}{34}}
                              {{13}{23}}
                              {{2}{122}}
                              {{3}{123}}
Inequivalent representatives of the a(4) = 19 matrices:
  [4] [2 2] [1 3]
.
  [1] [1 0] [1 0] [0 1] [2] [2 0] [1 1] [1 1]
  [3] [1 2] [0 3] [1 2] [2] [0 2] [1 1] [0 2]
.
  [1] [1 0] [1 0] [1 0] [0 1]
  [1] [1 0] [0 1] [0 1] [0 1]
  [2] [0 2] [1 1] [0 2] [1 1]
.
  [1] [1 0] [1 0]
  [1] [1 0] [0 1]
  [1] [0 1] [0 1]
  [1] [0 1] [0 1]

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

Cf. A007716, A052847, A005380, A054974, A005986, A120733, A316980, A323654, A323655, A323656.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={concat(1, (EulerT(vector(n, k, k+1)) + EulerT(vector(n, k, if(k%2, 0, (k+6)\4))))/2)} \\ Andrew Howroyd, Aug 26 2019

a(2*n) = (A005380(2*n) + A005986(n))/2; a(2*n+1) = A005380(2*n+1)/2. - Andrew Howroyd, Aug 26 2019

Terms a(11) and beyond from Andrew Howroyd, Aug 26 2019

A323656 Number of non-isomorphic multiset partitions of weight n with exactly 2 distinct vertices, or with exactly 2 (not necessarily distinct) edges.

Original entry on oeis.org

0, 0, 2, 4, 14, 28, 69, 134, 285, 536, 1050, 1918, 3566, 6346, 11363, 19771, 34405, 58677, 99797, 167223, 279032, 460264, 755560, 1228849, 1988680, 3193513, 5103104, 8100712, 12798207, 20102883, 31434374, 48900337, 75746745, 116787611, 179342230, 274238159

Offset: 0

Gus Wiseman, Jan 22 2019

nonn

The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

Also the number of nonnegative integer matrices with only two columns, no zero rows or columns, and sum of entries equal to n, up to row and column permutations.

			Non-isomorphic representatives of the a(2) = 2 through a(4) = 14 multiset partitions with exactly 2 distinct vertices:
  {{12}}    {{122}}      {{1122}}
  {{1}{2}}  {{1}{22}}    {{1222}}
            {{2}{12}}    {{1}{122}}
            {{1}{2}{2}}  {{11}{22}}
                         {{12}{12}}
                         {{1}{222}}
                         {{12}{22}}
                         {{2}{122}}
                         {{1}{1}{22}}
                         {{1}{2}{12}}
                         {{1}{2}{22}}
                         {{2}{2}{12}}
                         {{1}{1}{2}{2}}
                         {{1}{2}{2}{2}}
Non-isomorphic representatives of the a(2) = 2 through a(4) = 14 multiset partitions with exactly 2 edges:
  {{1}{1}}  {{1}{11}}  {{1}{111}}
  {{1}{2}}  {{1}{22}}  {{11}{11}}
            {{1}{23}}  {{1}{122}}
            {{2}{12}}  {{11}{22}}
                       {{12}{12}}
                       {{1}{222}}
                       {{12}{22}}
                       {{1}{233}}
                       {{12}{33}}
                       {{1}{234}}
                       {{12}{34}}
                       {{13}{23}}
                       {{2}{122}}
                       {{3}{123}}
Inequivalent representatives of the a(4) = 14 matrices:
  [2 2] [1 3]
.
  [1 0] [1 0] [0 1] [2 0] [1 1] [1 1]
  [1 2] [0 3] [1 2] [0 2] [1 1] [0 2]
.
  [1 0] [1 0] [1 0] [0 1]
  [1 0] [0 1] [0 1] [0 1]
  [0 2] [1 1] [0 2] [1 1]
.
  [1 0] [1 0]
  [1 0] [0 1]
  [0 1] [0 1]
  [0 1] [0 1]

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

Cf. A000041, A007716, A052847, A054974, A120733, A316980, A323654, A323655, A323656.

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={concat(0, (EulerT(vector(n, k, k+1)) + EulerT(vector(n, k, if(k%2, 0, (k+6)\4))))/2 - EulerT(vector(n,k,1)))} \\ Andrew Howroyd, Aug 26 2019

a(n) = A323655(n) - A000041(n). - Andrew Howroyd, Aug 26 2019

Terms a(11) and beyond from Andrew Howroyd, Aug 26 2019

cp's OEIS Frontend

A052847 G.f.: 1 / Product_{k>=1} (1-x^k)^(k-1).

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A054974 Number of nonnegative integer 2 X 2 matrices with no zero rows or columns and with sum of elements equal to n, up to row and column permutation.

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A323655 Number of non-isomorphic multiset partitions of weight n with at most 2 distinct vertices, or with at most 2 (not necessarily distinct) edges.

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A323656 Number of non-isomorphic multiset partitions of weight n with exactly 2 distinct vertices, or with exactly 2 (not necessarily distinct) edges.

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