A052847 -id:A052847 - cp's OEIS frontend

A263361 Expansion of Product_{k>=1} 1/(1-x^(k+5))^k.

1, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 8, 10, 15, 20, 30, 40, 58, 76, 106, 140, 191, 252, 344, 454, 613, 814, 1091, 1442, 1926, 2538, 3368, 4432, 5852, 7678, 10107, 13222, 17337, 22636, 29582, 38518, 50195, 65198, 84712, 109784, 142254, 183924, 237742, 306688

Offset: 0

Vaclav Kotesovec, Oct 16 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Cf. A000219, A052847, A263358, A263359, A263360, A263362, A263363, A263364.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      max(0, d-5), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Oct 16 2015

nmax = 60; CoefficientList[Series[Product[1/(1-x^(k+5))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[E^Sum[x^(6*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} x^(6*k)/(k*(1-x^k)^2)).

a(n) ~ exp(1/12 - 25*Pi^4/(432*Zeta(3)) - 5*Pi^2 * n^(1/3) / (3 * 2^(4/3) * Zeta(3)^(1/3)) + 3 * 2^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * n^(125/36) * Pi^2 / (576 * A * 2^(35/36) * sqrt(3) * Zeta(3)^(143/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A263362 Expansion of Product_{k>=1} 1/(1-x^(k+6))^k.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 9, 11, 16, 21, 31, 41, 58, 76, 103, 133, 178, 229, 303, 394, 519, 675, 889, 1155, 1513, 1964, 2558, 3310, 4298, 5543, 7169, 9231, 11903, 15289, 19665, 25208, 32339, 41374, 52943, 67595, 86307, 109965, 140089, 178155

Offset: 0

Vaclav Kotesovec, Oct 16 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Cf. A000219, A052847, A263358, A263359, A263360, A263361, A263363, A263364.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      max(0, d-6), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 16 2015

nmax = 60; CoefficientList[Series[Product[1/(1-x^(k+6))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[E^Sum[x^(7*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} x^(7*k)/(k*(1-x^k)^2)).

a(n) ~ exp(1/12 - Pi^4/(12*Zeta(3)) - Pi^2 * n^(1/3) / (2^(1/3) * Zeta(3)^(1/3)) + 3 * 2^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * n^(191/36) * Pi^(5/2) / (276480 * A * 2^(11/36) * sqrt(3) * Zeta(3)^(209/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A263363 Expansion of Product_{k>=1} 1/(1-x^(k+7))^k.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 17, 22, 32, 42, 59, 76, 103, 130, 171, 216, 280, 354, 460, 584, 757, 968, 1249, 1596, 2056, 2618, 3354, 4266, 5441, 6900, 8778, 11108, 14094, 17814, 22546, 28450, 35946, 45280, 57088, 71806, 90347

Offset: 0

Vaclav Kotesovec, Oct 16 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015

Cf. A000219, A052847, A263358, A263359, A263360, A263361, A263362, A263364.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      max(0, d-7), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 16 2015

nmax = 60; CoefficientList[Series[Product[1/(1-x^(k+7))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[E^Sum[x^(8*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} x^(8*k)/(k*(1-x^k)^2)).

a(n) ~ exp(1/12 - 49*Pi^4/(432*Zeta(3)) - 7*Pi^2 * n^(1/3) / (3 * 2^(4/3) * Zeta(3)^(1/3)) + 3 * 2^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * n^(269/36) * Pi^3 / (398131200 * A * 2^(35/36) * sqrt(3) * Zeta(3)^(287/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A263364 Expansion of Product_{k>=1} 1/(1-x^(k+8))^k.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 18, 23, 33, 43, 60, 77, 103, 130, 168, 209, 267, 331, 420, 526, 667, 839, 1069, 1347, 1711, 2160, 2733, 3437, 4336, 5435, 6828, 8543, 10699, 13357, 16703, 20820, 25986, 32362, 40327, 50152, 62407

Offset: 0

Vaclav Kotesovec, Oct 16 2015

nonn

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.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
Eric Weisstein's World of Mathematics, Barnes G-Function.

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).

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      max(0, d-8), d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 16 2015

nmax = 60; CoefficientList[Series[Product[1/(1-x^(k+8))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 60; CoefficientList[Series[E^Sum[x^(9*k)/(k*(1-x^k)^2), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: exp(Sum_{k>=1} x^(9*k)/(k*(1-x^k)^2)).

a(n) ~ exp(1/12 - 4*Pi^4/(27*Zeta(3)) - 2^(5/3) * Pi^2 * n^(1/3) / (3 * Zeta(3)^(1/3)) + 3 * 2^(-2/3) * Zeta(3)^(1/3) * n^(2/3)) * n^(359/36) * Pi^(7/2) / (8026324992000 * A * 2^(35/36) * sqrt(3) * Zeta(3)^(377/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A363601 Number of partitions of n where there are k^2 - 1 kinds of parts k.

Original entry on oeis.org

1, 0, 3, 8, 21, 48, 126, 288, 693, 1568, 3570, 7896, 17417, 37632, 80823, 171192, 359733, 747936, 1543192, 3155760, 6407037, 12909024, 25835649, 51359136, 101470854, 199264128, 389096028, 755591256, 1459643343, 2805471984, 5366161740, 10216161336, 19362398580

Offset: 0

Seiichi Manyama, Jun 10 2023

Cf. A005380, A023871, A052847, A092348, A120844, A217093, A253289, A255271, A255802, A255803, A255835, A255836, A363602.

with(numtheory):
series(exp(add((sigma[3](k) - sigma[1](k))*x^k/k, k = 1..50)), x, 51):
seq(coeftayl(%, x = 0, n), n = 0..50); # Peter Bala, Jan 16 2025

my(N=40, x='x+O('x^N)); Vec(1/prod(k=1, N, (1-x^k)^(k^2-1)))

G.f.: 1/Product_{k>=1} (1-x^k)^(k^2-1).
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} A092348(k) * a(n-k).
G.f.: exp(Sum_{k >= 1} (sigma_3(k) - sigma_1(k))*x^k/k) = 1 + 3*x^2 + 8*x^3 + 21*x^4 + 48*x^5 + .... - Peter Bala, Jan 16 2025

Name suggested by Joerg Arndt, Jun 11 2023

A263352 Expansion of Product_{k>=1} 1/(1 - x^(2*k+3))^k.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 0, 3, 1, 4, 2, 5, 6, 7, 10, 9, 19, 14, 29, 23, 46, 38, 66, 64, 99, 107, 143, 171, 211, 270, 311, 418, 465, 633, 698, 945, 1049, 1399, 1579, 2052, 2364, 2997, 3527, 4366, 5219, 6339, 7686, 9197, 11234, 13321, 16340, 19261, 23622, 27796

Offset: 0

Vaclav Kotesovec, Oct 16 2015

nonn

From Vaclav Kotesovec, Oct 17 2015: (Start)
In general, if g.f. = Product_{k>=1} 1/(1-x^(2*k+v))^k and v>0 is odd, then a(n) ~ d2(v) * (2*n)^(v^2/24 - 25/36) * exp(-Pi^4 * v^2 / (1728*Zeta(3)) - Pi^2 * v * n^(1/3) /(3 * 2^(8/3) * Zeta(3)^(1/3)) + 3*Zeta(3)^(1/3) * n^(2/3) / 2^(4/3)) / (sqrt(3*Pi) * Zeta(3)^(v^2/24 - 7/36)), where Zeta(3) = A002117.
d2(v) = exp(Integral_{x=0..infinity} 1/(x*exp((v-2)*x) * (exp(2*x)- 1)^2) - (3*v^2-2)/(24*x*exp(x)) + v/(4*x^2) - 1/(4*x^3) dx).
d2(v) = 2^(v/4 - 1/12) * exp(-Zeta'(-1)/2) / Product_{j=1..(v-1)/2} (2*j-1)!!, where Zeta'(-1) = A084448 and Product_{j=1..(v-1)/2} (2*j-1)!! = A057863((v-1)/2).
d2(v) = 2^(1/12 + v/4 - v^2/8) * exp(1/12) * Pi^(v/4) / (A * G(v/2 + 1)), where A = A074962 is the Glaisher-Kinkelin constant and G is the Barnes G-function.
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
Eric Weisstein's World of Mathematics, Barnes G-Function.

Cf. A052847, A000219, A263150, A035528, A263395, A263396, A263397.

with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(add(d*
      `if`(d>1 and d::odd, (d-3)/2, 0),
      d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Oct 17 2015

nmax = 60; CoefficientList[Series[Product[1/(1 - x^(2*k+3))^k, {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ 2^(25/72) * sqrt(A) * exp(-1/24 + 3 * 2^(-4/3) * Zeta(3)^(1/3) * n^(2/3) - Pi^4/(192*Zeta(3)) - Pi^2 * n^(1/3)/(2^(8/3) * Zeta(3)^(1/3))) / (sqrt(3*Pi) * Zeta(3)^(13/72) * n^(23/72)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A323654 Number of non-isomorphic multiset partitions of weight n with no constant parts and only two distinct vertices.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 8, 9, 20, 26, 50, 69, 125, 177, 301, 440, 717, 1055, 1675, 2471, 3835, 5660, 8627, 12697, 19095, 27978, 41581, 60650, 89244, 129490, 188925, 272676, 394809, 566882, 815191, 1164510, 1664295, 2365698, 3361844, 4756030, 6723280, 9468138, 13319299

Offset: 0

Gus Wiseman, Jan 22 2019

nonn

First differs from A304967 at a(10) = 50, A304967(10) = 49.
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 positive integer matrices with only two columns and sum of entries equal to n, up to row and column permutations.

			Non-isomorphic representatives of the a(2) = 1 through a(7) = 9 multiset partitions:
  {{12}}  {{122}}  {{1122}}    {{11222}}    {{111222}}      {{1112222}}
                   {{1222}}    {{12222}}    {{112222}}      {{1122222}}
                   {{12}{12}}  {{12}{122}}  {{122222}}      {{1222222}}
                                            {{112}{122}}    {{112}{1222}}
                                            {{12}{1122}}    {{12}{11222}}
                                            {{12}{1222}}    {{12}{12222}}
                                            {{122}{122}}    {{122}{1122}}
                                            {{12}{12}{12}}  {{122}{1222}}
                                                            {{12}{12}{122}}
Inequivalent representatives of the a(8) = 20 matrices:
  [4 4] [3 5] [2 6] [1 7]
.
  [1 1] [1 1] [1 1] [2 1] [2 1] [1 2] [1 2] [3 1] [2 2] [2 2] [1 3]
  [3 3] [2 4] [1 5] [2 3] [1 4] [2 3] [1 4] [1 3] [2 2] [1 3] [1 3]
.
  [1 1] [1 1] [1 1] [1 1]
  [1 1] [1 1] [2 1] [1 2]
  [2 2] [1 3] [1 2] [1 2]
.
  [1 1]
  [1 1]
  [1 1]
  [1 1]

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

Cf. A003293, A007716, A052847, A054974, A120733, A321407, A321760, 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+2)\4))))/2)} \\ Andrew Howroyd, Aug 26 2019

a(2*n) = (A052847(2*n) + A003293(n))/2; a(2*n+1) = A052847(2*n+1)/2. - Andrew Howroyd, Aug 26 2019

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

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

A264923 G.f.: 1 / Product_{n>=0} (1 - x^(n+3))^((n+1)*(n+2)/2!).

Original entry on oeis.org

1, 0, 0, 1, 3, 6, 11, 18, 33, 57, 105, 183, 330, 567, 990, 1693, 2904, 4917, 8343, 14010, 23511, 39171, 65100, 107592, 177352, 290931, 475905, 775381, 1259637, 2039094, 3291613, 5296467, 8499339, 13599292, 21702795, 34541724, 54839894, 86847255, 137212197, 216274466, 340129773, 533726442, 835732774, 1305877914, 2036369010

Offset: 0

Paul D. Hanna, Nov 28 2015

nonn

Number of partitions of n objects of 3 colors, where each part must contain at least one of each color. [Conjecture - see comment by Franklin T. Adams-Watters in A052847].

			G.f.: A(x) = 1 + x^3 + 3*x^4 + 6*x^5 + 11*x^6 + 18*x^7 + 33*x^8 + 57*x^9 + 105*x^10 +...
where
1/A(x) = (1-x^3) * (1-x^4)^3 * (1-x^5)^6 * (1-x^6)^10 * (1-x^7)^15 * (1-x^8)^21 * (1-x^9)^28 * (1-x^10)^36 * (1-x^11)^45 *...
Also,
log(A(x)) = (x/(1-x))^3 + (x^2/(1-x^2))^3/2 + (x^3/(1-x^3))^3/3 + (x^4/(1-x^4))^3/4 + (x^5/(1-x^5))^3/5 + (x^6/(1-x^6))^3/6 +...

Vaclav Kotesovec, Table of n, a(n) for n = 0..6000

Cf. A052847, A264924, A264925, A264926.

Cf. A000294, A217093, A258349.

nmax = 50; CoefficientList[Series[Product[1/(1-x^k)^((k-2)*(k-1)/2), {k,1,nmax}], {x,0,nmax}], x] (* Vaclav Kotesovec, Dec 09 2015 *)

{a(n) = my(A=1); A = prod(k=0,n, 1/(1 - x^(k+3) +x*O(x^n) )^((k+1)*(k+2)/2) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

{a(n) = my(A=1); A = exp( sum(k=1,n+1, (x^k/(1 - x^k))^3 /k +x*O(x^n) ) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

{L(n) = sumdiv(n,d, d*(d-1)*(d-2)/2! )}
{a(n) = my(A=1); A = exp( sum(k=1,n+1, L(k) * x^k/k +x*O(x^n) ) ); polcoeff(A,n)}
for(n=0,50,print1(a(n),", "))

G.f.: exp( Sum_{n>=1} ( x^n/(1-x^n) )^3 /n ).
G.f.: exp( Sum_{n>=1} L(n) * x^n/n ), where L(n) = Sum_{d|n} d*(d-1)*(d-2)/2!.
a(n) ~ Pi^(3/8) / (2^(55/32) * 15^(7/32) * n^(23/32)) * exp(29*Zeta(3)/(8*Pi^2) - log(2*Pi)/2 - 3*Zeta'(-1)/2 - 2025*Zeta(3)^3/(2*Pi^8) + (5^(1/4)*Pi/6^(3/4) - 135*15^(1/4)*Zeta(3)^2/(2^(7/4)*Pi^5)) * n^(1/4) - 3*sqrt(15*n/2)*Zeta(3)/Pi^2 + 2^(7/4)*Pi/(3*15^(1/4)) * n^(3/4)). - Vaclav Kotesovec, Dec 09 2015

cp's OEIS Frontend

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A323654 Number of non-isomorphic multiset partitions of weight n with no constant parts and only two distinct vertices.

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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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