A363045 -id:A363045 - cp's OEIS frontend

A363048 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) is the number of partitions of n whose greatest part is a multiple of k.

1, 0, 1, 0, 2, 1, 0, 3, 1, 1, 0, 5, 3, 1, 1, 0, 7, 3, 2, 1, 1, 0, 11, 6, 4, 2, 1, 1, 0, 15, 7, 5, 3, 2, 1, 1, 0, 22, 12, 7, 6, 3, 2, 1, 1, 0, 30, 14, 11, 7, 5, 3, 2, 1, 1, 0, 42, 22, 14, 11, 8, 5, 3, 2, 1, 1, 0, 56, 27, 19, 14, 11, 7, 5, 3, 2, 1, 1, 0, 77, 40, 27, 21, 15, 12, 7, 5, 3, 2, 1, 1

Offset: 0

Seiichi Manyama, May 14 2023

			Triangle begins:
  1;
  0,  1;
  0,  2,  1;
  0,  3,  1,  1;
  0,  5,  3,  1, 1;
  0,  7,  3,  2, 1, 1;
  0, 11,  6,  4, 2, 1, 1;
  0, 15,  7,  5, 3, 2, 1, 1;
  0, 22, 12,  7, 6, 3, 2, 1, 1;
  0, 30, 14, 11, 7, 5, 3, 2, 1, 1;
  ...

Alois P. Heinz, Rows n = 0..200, flattened

Row sums give A323433.
Column k=0..5 give A000007, A000041, A027187, A363045, A363046, A363047.
T(2n,n) gives A052810.
Cf. A008284, A072233, A350890.

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-1)+b(n-i, min(n-i, i))))
    end:
T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), add(
    (j-> b(n-j, min(n-j, j)))(k*i), i=0..n/k)):
seq(seq(T(n, k), k=0..n), n=0..12);  # Alois P. Heinz, May 14 2023

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, b[n, i - 1] + b[n - i, Min[n - i, i]]]];
T[n_, k_] := If[k == 0, If[n == 0, 1, 0], Sum[Function[j, b[n - j, Min[n - j, j]]][k*i], {i, 0, n/k}]];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Oct 20 2023, after Alois P. Heinz *)

T(n, k) = sum(j=0, n, #partitions(n-k*j, k*j));

For k > 0, g.f. of column k: Sum_{i>=0} x^(k*i)/Product_{j=1..k*i} (1-x^j).

A363066 Number of partitions p of n such that (1/3)*max(p) is a part of p.

Original entry on oeis.org

1, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9, 11, 16, 20, 27, 33, 45, 55, 72, 89, 116, 142, 181, 222, 281, 343, 429, 522, 649, 786, 967, 1168, 1429, 1719, 2088, 2504, 3026, 3615, 4345, 5174, 6192, 7349, 8755, 10360, 12297, 14507, 17154, 20182, 23788, 27910, 32790, 38374, 44955, 52480, 61307, 71402

Offset: 0

Seiichi Manyama, May 16 2023

nonn

			a(7) = 3 counts these partitions:  331, 3211, 31111.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)

Cf. A002865, A238479, A363067, A363068.
Cf. A008484, A237825, A363045.
Cf. A035295.

nmax = 60; CoefficientList[Series[Sum[x^(4*k)/Product[1 - x^j, {j, 1, 3*k}], {k, 0, nmax}], {x, 0, nmax}], x]  (* Vaclav Kotesovec, Jun 18 2025 *)
nmax = 60; p=1; s=1; Do[p=Expand[p*(1-x^(3*k))*(1-x^(3*k-1))*(1-x^(3*k-2))]; p=Take[p, Min[nmax+1, Exponent[p, x]+1, Length[p]]]; s+=x^(4*k)/p; , {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 18 2025 *)
Join[{1},Table[Count[IntegerPartitions[n],?(MemberQ[#,#[[1]]/3]&)],{n,60}]] (* _Harvey P. Dale, Jun 29 2025 *)

a(n) = sum(k=0, n\4, #partitions(n-4*k, 3*k));

G.f.: Sum_{k>=0} x^(4*k)/Product_{j=1..3*k} (1-x^j).

a(n) ~ Gamma(1/3) * Pi^(1/3) * exp(Pi*sqrt(2*n/3)) / (2^(13/6) * 3^(8/3) * n^(7/6)). - Vaclav Kotesovec, Jun 19 2025

A363094 Number of partitions of n whose least part is a multiple of 3.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 1, 1, 3, 2, 3, 6, 6, 7, 11, 11, 14, 21, 24, 29, 38, 44, 54, 69, 81, 98, 123, 144, 174, 213, 253, 300, 363, 427, 508, 608, 716, 846, 1004, 1176, 1384, 1631, 1908, 2230, 2616, 3046, 3553, 4143, 4813, 5586, 6492, 7509, 8693, 10057, 11608, 13383, 15435, 17753, 20418, 23463, 26923, 30864

Offset: 1

Seiichi Manyama, May 19 2023

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A026805, A363095, A363096.

Cf. A363045.

nmax = 60; Rest[CoefficientList[Series[Sum[x^(3*k)/QPochhammer[x^(3*k), x], {k, 1, nmax/3}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, May 20 2023 *)

my(N=70, x='x+O('x^N)); concat([0, 0], Vec(sum(k=1, N, x^(3*k)/prod(j=3*k, N, 1-x^j))))

G.f.: Sum_{k>=1} x^(3*k)/Product_{j>=3*k} (1-x^j).

a(n) ~ Pi^2 * exp(Pi*sqrt(2*n/3)) / (4 * 3^(3/2) * n^2) * (1 - (3*sqrt(6)/Pi + 109*Pi*sqrt(6)/144)/sqrt(n)). - Vaclav Kotesovec, May 21 2023

A372893 Number of distinct partitions p of n such that max(p) == 0 mod 3.

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 2, 1, 1, 3, 3, 4, 6, 6, 7, 10, 11, 12, 16, 17, 20, 26, 29, 34, 42, 47, 54, 66, 74, 85, 101, 113, 129, 151, 170, 193, 224, 252, 286, 329, 370, 418, 478, 536, 603, 686, 767, 862, 974, 1088, 1218, 1370, 1527, 1704, 1910, 2124, 2366, 2643, 2934, 3260, 3631

Offset: 0

Seiichi Manyama, May 20 2024

nonn

			a(9) = 3 counts these partitions: 9, 63, 621.

Column 3 of A373029.
Cf. A000009, A373012, A373013.
Cf. A026838, A363045.

G.f.: Sum_{k>=0} x^(3*k) * Product_{j=1..3*k-1} (1+x^j).

A000009(n) = a(n) + A373012(n) + A373013(n).

A370747 Number of partitions of n into distinct parts such that number of parts is a multiples of 3.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 1, 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 28, 31, 35, 40, 45, 51, 59, 66, 76, 87, 100, 114, 133, 151, 175, 201, 232, 265, 307, 349, 402, 458, 524, 594, 680, 767, 872, 983, 1112, 1248, 1409, 1575, 1770, 1976, 2211, 2460, 2748, 3048, 3393, 3759, 4173, 4612, 5112

Offset: 0

Seiichi Manyama, May 23 2024

nonn

			a(12) = 7 counts these partitions: 921, 831, 741, 732, 651, 642, 543.

Cf. A000009, A067661, A372703, A373078.

Cf. A363045.

my(N=70, x='x+O('x^N)); Vec(sum(k=0, N, prod(j=1, 3*k, x^j/(1-x^j))))

G.f.: Sum_{k>=0} x^(3*k*(3*k+1)/2) / Product_{j=1..3*k} (1-x^j) = Sum_{k>=0} Product_{j=1..3*k} (x^j/(1-x^j)).

A373014 Number of partitions p of n such that max(p) == 1 mod 3.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 5, 7, 9, 14, 18, 25, 34, 45, 58, 78, 99, 128, 165, 210, 264, 336, 419, 525, 655, 813, 1003, 1242, 1522, 1867, 2283, 2783, 3379, 4105, 4960, 5989, 7214, 8670, 10391, 12447, 14858, 17719, 21088, 25055, 29705, 35187, 41581, 49084, 57844, 68072, 79974

Offset: 0

Seiichi Manyama, May 20 2024

nonn

			a(7) = 5 counts these partitions: 7, 43, 421, 4111, 1111111.

Cf. A000041, A363045, A373015.

Cf. A027193.

my(N=60, x='x+O('x^N)); concat(0, Vec(sum(k=0, N, x^(3*k+1)/prod(j=1, 3*k+1, 1-x^j))))

G.f.: Sum_{k>=0} x^(3*k+1) / Product_{j=1..3*k+1} (1-x^j).

A000041(n) = A363045(n) + a(n) + A373015(n).

A373015 Number of partitions p of n such that max(p) == 2 mod 3.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 4, 5, 8, 10, 14, 19, 25, 33, 45, 58, 76, 99, 127, 162, 209, 263, 333, 419, 524, 652, 813, 1003, 1239, 1524, 1868, 2281, 2786, 3382, 4104, 4965, 5993, 7213, 8676, 10396, 12447, 14866, 17725, 21087, 25063, 29711, 35185, 41589, 49089, 57839, 68079

Offset: 0

Seiichi Manyama, May 20 2024

nonn

			a(8) = 8 counts these partitions: 8, 53, 521, 5111, 2222, 22211, 221111, 2111111.

Cf. A000041, A363045, A373014.

my(N=60, x='x+O('x^N)); concat([0, 0], Vec(sum(k=0, N, x^(3*k+2)/prod(j=1, 3*k+2, 1-x^j))))

G.f.: Sum_{k>=0} x^(3*k+2) / Product_{j=1..3*k+2} (1-x^j).

A000041(n) = A363045(n) + A373014(n) + a(n).

cp's OEIS Frontend

A363048 Triangle T(n,k), n >= 0, 0 <= k <= n, read by rows, where T(n,k) is the number of partitions of n whose greatest part is a multiple of k.

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A363066 Number of partitions p of n such that (1/3)*max(p) is a part of p.

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A363094 Number of partitions of n whose least part is a multiple of 3.

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A372893 Number of distinct partitions p of n such that max(p) == 0 mod 3.

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A370747 Number of partitions of n into distinct parts such that number of parts is a multiples of 3.

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A373014 Number of partitions p of n such that max(p) == 1 mod 3.

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A373015 Number of partitions p of n such that max(p) == 2 mod 3.

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