A363046 -id:A363046 - 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).

A363067 Number of partitions p of n such that (1/4)*max(p) is a part of p.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 31, 39, 51, 64, 81, 102, 128, 159, 198, 245, 304, 374, 460, 563, 689, 841, 1023, 1242, 1505, 1819, 2195, 2642, 3173, 3804, 4551, 5435, 6477, 7707, 9151, 10850, 12843, 15175, 17902, 21089, 24802, 29132, 34164, 40012, 46796, 54663, 63766

Offset: 0

Seiichi Manyama, May 16 2023

nonn

			a(8) = 3 counts these partitions:  431, 4211, 41111.

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

Cf. A002865, A238479, A363066, A363068.
Cf. A237826, A363046.
Cf. A035296.

nmax = 60; CoefficientList[Series[Sum[x^(5*k)/Product[1 - x^j, {j, 1, 4*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^(4*k))*(1-x^(4*k-1))*(1-x^(4*k-2))*(1-x^(4*k-3))]; p=Take[p, Min[nmax+1, Exponent[p, x]+1, Length[p]]]; s+=x^(5*k)/p; , {k, 1, nmax}]; CoefficientList[Series[s, {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 18 2025 *)

a(n) = sum(k=0, n\5, #partitions(n-5*k, 4*k));

G.f.: Sum_{k>=0} x^(5*k)/Product_{j=1..4*k} (1-x^j).

a(n) ~ Gamma(1/4) * Pi^(1/4) * exp(Pi*sqrt(2*n/3)) / (2^(49/8) * 3^(5/8) * n^(9/8)). - Vaclav Kotesovec, Jun 19 2025

A363095 Number of partitions of n whose least part is a multiple of 4.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 1, 1, 3, 2, 3, 3, 7, 6, 8, 9, 13, 13, 17, 19, 28, 30, 38, 43, 56, 62, 76, 87, 110, 124, 151, 173, 211, 241, 289, 332, 399, 456, 539, 620, 733, 838, 983, 1127, 1322, 1513, 1761, 2016, 2343, 2677, 3096, 3536, 4083, 4655, 5355, 6101, 7005, 7969, 9124, 10370, 11856, 13453, 15340

Offset: 1

Seiichi Manyama, May 19 2023

nonn

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

Cf. A026805, A363094, A363096.

Cf. A363046.

nmax = 60; Rest[CoefficientList[Series[Sum[x^(4*k)/QPochhammer[x^(4*k), x], {k, 1, nmax/4}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, May 20 2023 *)

my(N=70, x='x+O('x^N)); concat([0, 0, 0], Vec(sum(k=1, N, x^(4*k)/prod(j=4*k, N, 1-x^j))))

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

a(n) ~ Pi^3 * exp(Pi*sqrt(2*n/3)) / (3 * 2^(5/2) * n^(5/2)) * (1 - (5*sqrt(6)/Pi + 169*Pi*sqrt(6)/144)/sqrt(n)). - Vaclav Kotesovec, May 21 2023

A372703 Number of partitions of n into distinct parts such that number of parts is a multiples of 4.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 6, 9, 11, 15, 18, 23, 27, 34, 39, 47, 54, 64, 72, 84, 94, 108, 120, 136, 150, 169, 185, 207, 226, 251, 273, 302, 328, 362, 393, 433, 470, 518, 563, 621, 677, 748, 818, 906, 994, 1104, 1216, 1354, 1497, 1671, 1853, 2073, 2305, 2582, 2877, 3226, 3599

Offset: 0

Seiichi Manyama, May 23 2024

nonn

Cf. A000009, A067661, A370747, A373078.

Cf. A363046.

my(N=70, x='x+O('x^N)); Vec(sum(k=0, N, prod(j=1, 4*k, x^j/(1-x^j))))

G.f.: Sum_{k>=0} x^(2*k*(4*k+1)) / Product_{j=1..4*k} (1-x^j) = Sum_{k>=0} Product_{j=1..4*k} (x^j/(1-x^j)).

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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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A363067 Number of partitions p of n such that (1/4)*max(p) is a part of p.

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A363095 Number of partitions of n whose least part is a multiple of 4.

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A372703 Number of partitions of n into distinct parts such that number of parts is a multiples of 4.

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