A280204 -id:A280204 - cp's OEIS frontend

A117144 Partitions of n in which each part k occurs at least k times.

1, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 8, 9, 10, 12, 15, 16, 19, 21, 25, 28, 32, 34, 41, 46, 51, 55, 64, 70, 79, 86, 97, 106, 119, 129, 146, 159, 175, 190, 214, 232, 256, 277, 306, 334, 367, 394, 434, 472, 515, 556, 607, 654, 714, 770, 836, 901, 978, 1048, 1140, 1226, 1322

Offset: 0

Emeric Deutsch, Mar 06 2006

nonn

The Heinz numbers of these integer partitions are given by A324525. - Gus Wiseman, Mar 09 2019

			a(9)=5 because we have [3,3,3], [2,2,2,2,1], [2,2,2,1,1,1], [2,2,1,1,1,1,1] and [1,1,1,1,1,1,1,1,1].
From _Gus Wiseman_, Mar 09 2019: (Start)
The a(1) = 1 through a(9) = 5 integer partitions:
  1  11  111  22    221    222     2221     2222      333
              1111  11111  2211    22111    22211     22221
                           111111  1111111  221111    222111
                                            11111111  2211111
                                                      111111111
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A001462, A003114, A006141, A033461, A039900, A047993, A052335, A064174, A090858, A114638, A115584, A276078, A280204.

Cf. A324518, A324520, A324525, A324572.

g:=product((1-x^k+x^(k^2))/(1-x^k),k=1..100): gser:=series(g,x=0,70): seq(coeff(gser,x,n),n=0..66);
# second Maple program:
b:= proc(n,i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1) +add(b(n-i*j, i-1), j=i..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Dec 28 2016

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + Sum[b[n-i*j, i-1], {j, i, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Feb 03 2017, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],And@@Table[Count[#,i]>=i,{i,Union[#]}]&]],{n,0,30}] (* Gus Wiseman, Mar 09 2019 *)
nmax = 100; CoefficientList[Series[Product[(1-x^k+x^(k^2))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 28 2024 *)

G.f.: Product_{k>=1} (1-x^k+x^(k^2))/(1-x^k).

A369520 Expansion of Product_{k>=1} 1/((1 - x^(k^2))*(1 - x^k)).

Original entry on oeis.org

1, 2, 4, 7, 13, 21, 34, 52, 80, 119, 175, 251, 359, 504, 702, 965, 1320, 1785, 2401, 3200, 4245, 5589, 7324, 9535, 12364, 15944, 20478, 26175, 33338, 42279, 53438, 67283, 84454, 105642, 131764, 163826, 203149, 251185, 309799, 381079, 467666, 572520, 699342, 852314

Offset: 0

Vaclav Kotesovec, Jan 25 2024

nonn

Convolution of A001156 and A000041.

a(n) is the number of pairs (Q(k), P(n-k)), 0<=k<=n, where Q(k) is a partition of k into squares and P(n-k) is a partition of n-k.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (250000 terms)

Cf. A000041, A001156, A087153, A264393, A280204, A369519, A369579.

nmax=50; CoefficientList[Series[Product[1/(1-x^(k^2))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(2*n/3) + 3^(1/4)*zeta(3/2)*n^(1/4)/2^(3/4) - 3*zeta(3/2)^2/(32*Pi)) / (2^(13/4) * 3^(3/4) * sqrt(Pi) * n^(5/4)).

A369570 Expansion of Product_{k>=1} (1 + x^(k^2)) * (1 + x^k).

Original entry on oeis.org

1, 2, 2, 3, 5, 7, 9, 12, 15, 20, 27, 33, 41, 52, 65, 80, 99, 120, 145, 177, 213, 255, 305, 363, 430, 511, 604, 709, 833, 976, 1141, 1331, 1547, 1793, 2079, 2406, 2775, 3197, 3676, 4221, 4841, 5541, 6330, 7225, 8235, 9372, 10655, 12094, 13710, 15529, 17568, 19848

Offset: 0

Vaclav Kotesovec, Jan 26 2024

nonn

Convolution of A033461 and A000009.

a(n) is the number of pairs (Q(k), P(n-k)), 0<=k<=n, where Q(k) is a partition of k into distinct squares and P(n-k) is a partition of n-k into distinct parts.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Cf. A000009, A033461, A280204, A280276.

nmax = 100; CoefficientList[Series[Product[(1+x^(k^2))*(1+x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(n/3) + 3^(1/4) * (sqrt(2) - 1) * zeta(3/2) * n^(1/4)/2 - 3*(3 - 2*sqrt(2)) * zeta(3/2)^2/(32*Pi)) / (2^(5/2) * 3^(1/4) * n^(3/4)).

A280278 G.f.: Product_{k>=1} (1 + x^(k^3)) / (1 - x^k).

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 18, 26, 38, 54, 75, 103, 141, 190, 254, 337, 444, 580, 754, 973, 1250, 1597, 2030, 2568, 3237, 4061, 5076, 6322, 7847, 9705, 11968, 14711, 18033, 22043, 26873, 32677, 39642, 47972, 57924, 69787, 83904, 100667, 120547, 144072, 171876, 204677

Offset: 0

Vaclav Kotesovec, Dec 30 2016

nonn

Convolution of A279329 and A000041.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000041, A264393, A279329, A280204, A369571, A369579.

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

a(n) ~ exp(Pi*sqrt(2*n/3) + (2^(1/3) - 1) * Gamma(1/3) * Zeta(4/3) * n^(1/6) / (2^(1/6) * 3^(5/6) * Pi^(1/3))) / (4*sqrt(6)*n).

A280224 G.f.: Product_{k>=1} (1 + 2*x^(k^2)) / (1-x^k).

Original entry on oeis.org

1, 3, 4, 7, 13, 23, 33, 51, 74, 110, 158, 222, 307, 429, 587, 792, 1065, 1423, 1877, 2470, 3227, 4196, 5416, 6963, 8899, 11344, 14384, 18158, 22840, 28647, 35786, 44552, 55295, 68423, 84422, 103879, 127457, 156009, 190481, 232007, 281934, 341879, 413640

Offset: 0

Vaclav Kotesovec, Dec 29 2016

nonn

Convolution of A279360 and A000041.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000041, A033461, A279360, A280204, A280225.

nmax=50; CoefficientList[Series[Product[(1+2*x^(k^2))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt((2*n)/3) + 3^(1/4)*c*n^(1/4)/ 2^(3/4) - 3*c^2/(32*Pi)) / (12*n), where c = -PolyLog(3/2, -2).

A280225 G.f.: Product_{k>=1} (1 + 3*x^(k^2)) / (1-x^k).

Original entry on oeis.org

1, 4, 5, 9, 17, 34, 47, 75, 109, 165, 240, 341, 473, 671, 936, 1268, 1722, 2325, 3091, 4099, 5403, 7083, 9207, 11923, 15339, 19682, 25134, 31909, 40378, 50954, 64068, 80171, 100089, 124506, 154465, 191043, 235636, 289816, 355673, 435285, 531486, 647478

Offset: 0

Vaclav Kotesovec, Dec 29 2016

nonn

Convolution of A279368 and A000041.

In general, if m >= 0 and g.f. = Product_{k>=1} (1 + m*x^(k^2)) / (1-x^k), then a(n) ~ exp(Pi*sqrt((2*n)/3) + 3^(1/4)*c*n^(1/4)/ 2^(3/4) - 3*c^2/(32*Pi)) / (4*sqrt(3)*sqrt(m+1)*n), where c = -PolyLog(3/2, -m).

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000041, A033461, A279368, A280204, A280224.

nmax=50; CoefficientList[Series[Product[(1+3*x^(k^2))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt((2*n)/3) + 3^(1/4)*c*n^(1/4)/ 2^(3/4) - 3*c^2/(32*Pi)) / (8*sqrt(3)*n), where c = -PolyLog(3/2, -3).

A280421 G.f.: Product_{k>=1} (1 + x^(k*(k+1)/2)) / (1 - x^k).

Original entry on oeis.org

1, 2, 3, 6, 10, 15, 24, 36, 52, 76, 109, 152, 211, 290, 393, 530, 709, 938, 1236, 1618, 2102, 2720, 3500, 4477, 5707, 7242, 9146, 11511, 14435, 18030, 22451, 27868, 34476, 42531, 52324, 64186, 78541, 95867, 116721, 141791, 171862, 207844, 250846, 302134

Offset: 0

Vaclav Kotesovec, Jan 02 2017

nonn

Convolution of A024940 and A000041.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A024940, A000041, A280204, A280423.

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

a(n) ~ exp(sqrt(2*n/3)*Pi + 3^(1/4) * (sqrt(2)-1) * Zeta(3/2) * n^(1/4) / 2^(3/4) + 3*(2*sqrt(2)-3) * Zeta(3/2)^2 / (32*Pi)) / (8*sqrt(3)*n).

A369573 Expansion of Product_{k>=1} (1 + x^(k^2)) / (1 - x^(k^3)).

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 4, 4, 5, 7, 8, 8, 9, 11, 12, 12, 14, 17, 18, 18, 20, 23, 24, 24, 26, 31, 34, 35, 38, 43, 46, 47, 50, 55, 59, 61, 66, 73, 77, 79, 84, 92, 97, 100, 106, 115, 121, 124, 130, 140, 148, 152, 161, 174, 183, 188, 197, 210, 220, 226, 235, 251, 264, 272

Offset: 0

Vaclav Kotesovec, Jan 26 2024

nonn

Convolution of A033461 and A003108.

a(n) is the number of pairs (Q(k), P(n-k)), 0<=k<=n, where Q(k) is a partition of k into distinct squares and P(n-k) is a partition of n-k into cubes.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A003108, A033461, A280204, A280277.

nmax = 100; CoefficientList[Series[Product[(1+x^(k^2))/(1-x^(k^3)), {k, 1, nmax^(1/2)}], {x, 0, nmax}], x]

cp's OEIS Frontend

A117144 Partitions of n in which each part k occurs at least k times.

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A369520 Expansion of Product_{k>=1} 1/((1 - x^(k^2))*(1 - x^k)).

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A369570 Expansion of Product_{k>=1} (1 + x^(k^2)) * (1 + x^k).

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A280278 G.f.: Product_{k>=1} (1 + x^(k^3)) / (1 - x^k).

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A280224 G.f.: Product_{k>=1} (1 + 2*x^(k^2)) / (1-x^k).

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A280225 G.f.: Product_{k>=1} (1 + 3*x^(k^2)) / (1-x^k).

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A280421 G.f.: Product_{k>=1} (1 + x^(k*(k+1)/2)) / (1 - x^k).

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A369573 Expansion of Product_{k>=1} (1 + x^(k^2)) / (1 - x^(k^3)).

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