A096403 -id:A096403 - cp's OEIS frontend

A188216 Expansion of 1 + Sum_{n>=1} (x^(n^2) / Product_{k>=n} (1 - x^k)).

1, 1, 1, 2, 4, 5, 8, 12, 17, 25, 34, 46, 64, 86, 114, 151, 200, 258, 335, 431, 552, 703, 891, 1121, 1411, 1764, 2196, 2725, 3374, 4155, 5111, 6260, 7650, 9319, 11329, 13726, 16608, 20031, 24114, 28962, 34725, 41529, 49595, 59095, 70304, 83476, 98968, 117109

Offset: 0

Joerg Arndt, Mar 24 2011

nonn

Number of partitions of n such that if k is the least part, then k occurs at least k times. - Joerg Arndt, Apr 17 2011

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

Cf. A096403 (expansion of sum(n>=1, x^(n^2) / prod(k>=n+1, 1-x^k)) ).

Cf. A003114 (largest part k occurs at least k times).

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
    end:
a:= n-> `if`(n=0, 1, add(b(n-j^2, j), j=1..isqrt(n))):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 03 2021

b[n_, i_] := b[n, i] = If[n==0, 1, If[i>n, 0, b[n, i+1] + b[n-i, i]]];
a[n_] := If[n==0, 1, Sum[b[n - j^2, j], {j, 1, Sqrt[n]}]];
a /@ Range[0, 50] (* Jean-François Alcover, Jan 25 2021, after Alois P. Heinz *)

N=55; x='x+O('x^N);
t=1+sum(n=1,N,x^(n^2)/prod(k=n,N,1-x^k));
Vec(t)

A240175 Number of partitions of n such that the least part is less than its multiplicity.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 6, 7, 12, 16, 24, 32, 47, 60, 84, 110, 148, 191, 254, 323, 423, 535, 687, 864, 1100, 1371, 1726, 2141, 2669, 3290, 4075, 4990, 6136, 7481, 9137, 11087, 13471, 16264, 19659, 23641, 28438, 34060, 40801, 48676, 58074, 69049, 82064, 97246

Offset: 0

Clark Kimberling, Apr 02 2014

			a(8) counts these 12 partitions:  611, 5111, 4211, 41111, 3311, 32111, 311111, 2222, 22211, 221111, 2111111, 11111111.

Cf. A188216, A096403, A240176, A240177.

z = 60; f[n_] := f[n] = IntegerPartitions[n];
Table[Count[f[n], p_ /; Min[p] < Count[p, Min[p]]], {n, 0, z}](* A240175 *)
Table[Count[f[n], p_ /; Min[p] <= Count[p, Min[p]]], {n, 0, z}](* A188216 *)
Table[Count[f[n], p_ /; Min[p] == Count[p, Min[p]]], {n, 0, z}](* A096403 *)
Table[Count[f[n], p_ /; Min[p] > Count[p, Min[p]]], {n, 0, z}] (* A240176 *)
Table[Count[f[n], p_ /; Min[p] >= Count[p, Min[p]]], {n, 0, z}] (* A240177 *)

a(n) = A188216(n) - A096403(n), for n >= 0.

A240176 Number of partitions of n such that (least part) > (multiplicity of least part).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 3, 3, 5, 5, 8, 10, 13, 15, 21, 25, 31, 39, 50, 59, 75, 89, 111, 134, 164, 194, 240, 285, 344, 410, 493, 582, 699, 824, 981, 1157, 1369, 1606, 1901, 2223, 2613, 3054, 3579, 4166, 4871, 5658, 6590, 7645, 8877, 10264, 11900, 13733, 15868

Offset: 0

Clark Kimberling, Apr 02 2014

			a(8) counts these 5 partitions:  8, 61, 53, 44, 332.

Cf. A188216, A096403, A240175, A240177.

z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Min[p] < Count[p, Min[p]]], {n, 0, z}]  (* A240175 *)
t2 = Table[Count[f[n], p_ /; Min[p] <= Count[p, Min[p]]], {n, 0, z}] (* A188216 *)
t3 = Table[Count[f[n], p_ /; Min[p] == Count[p, Min[p]]], {n, 0, z}] (* A096403 *)
t4 = Table[Count[f[n], p_ /; Min[p] > Count[p, Min[p]]], {n, 0, z}] (* A240176 *)
t5 = Table[Count[f[n], p_ /; Min[p] >= Count[p, Min[p]]], {n, 0, z}] (* A240177 *)

a(n) + A240175(n) + A096403(n) = A000041(n), for n >= 0.

A240177 Number of partitions of n such that (least part) >= (multiplicity of least part).

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 8, 10, 14, 18, 24, 30, 41, 51, 66, 83, 106, 131, 167, 204, 257, 315, 391, 475, 587, 710, 869, 1049, 1275, 1529, 1852, 2213, 2662, 3173, 3796, 4506, 5373, 6356, 7544, 8900, 10523, 12373, 14585, 17101, 20085, 23494, 27508, 32087, 37471

Offset: 0

Clark Kimberling, Apr 02 2014

			a(6) counts these 5 partitions:  6, 51, 42, 33, 321.

Cf. A188216, A096403, A240175, A240176.

z = 60; f[n_] := f[n] = IntegerPartitions[n]; t1 = Table[Count[f[n], p_ /; Min[p] < Count[p, Min[p]]], {n, 0, z}]  (* A240175 *)
t2 = Table[Count[f[n], p_ /; Min[p] <= Count[p, Min[p]]], {n, 0, z}] (* A188216 *)
t3 = Table[Count[f[n], p_ /; Min[p] == Count[p, Min[p]]], {n, 0, z}] (* A096403 *)
t4 = Table[Count[f[n], p_ /; Min[p] > Count[p, Min[p]]], {n, 0, z}] (* A240176 *)
t5 = Table[Count[f[n], p_ /; Min[p] >= Count[p, Min[p]]], {n, 0, z}] (* A240177 *)

a(n) = A096403(n) + A240176(n), for n >= 0.

cp's OEIS Frontend

A188216 Expansion of 1 + Sum_{n>=1} (x^(n^2) / Product_{k>=n} (1 - x^k)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A240175 Number of partitions of n such that the least part is less than its multiplicity.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A240176 Number of partitions of n such that (least part) > (multiplicity of least part).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A240177 Number of partitions of n such that (least part) >= (multiplicity of least part).

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula