A089333 -id:A089333 - cp's OEIS frontend

A103198 Number of compositions of n into a square number of parts.

1, 1, 1, 1, 2, 5, 11, 21, 36, 58, 94, 166, 331, 716, 1574, 3368, 6892, 13447, 25127, 45391, 80428, 142615, 259085, 491855, 982400, 2045001, 4352661, 9291361, 19609786, 40574017, 81973315, 161568281, 311062991, 586764281, 1089615033, 2005257849, 3688711427

Offset: 0

Vladeta Jovovic, Mar 18 2005

From Gus Wiseman, Jan 17 2019: (Start)
Also the number of ways to fill a square matrix with the parts of an integer partition of n. For example, the a(6) = 11 matrices are:
  [6]
.
  [1 1] [1 1] [1 3] [3 1] [1 1] [1 2] [1 2] [2 1] [2 1] [2 2]
  [1 3] [3 1] [1 1] [1 1] [2 2] [1 2] [2 1] [1 2] [2 1] [1 1]
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..3329 (terms n = 1..1000 from Vaclav Kotesovec)
Vaclav Kotesovec, a(n+1)/a(n) as a graph

Cf. A000290, A011782, A052467, A089299, A089333, A120732, A323433, A323519, A323525.

b:= proc(n, t) option remember; `if`(n=0,
      `if`(issqr(t), 1, 0), add(b(n-j, t+1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..40);  # Alois P. Heinz, Jan 18 2019

nmax = 40; Rest[CoefficientList[Series[-1/2 + EllipticTheta[3, 0, x/(1-x)]/2, {x, 0, nmax}], x]] (* Vaclav Kotesovec, Jan 03 2017 *)

a(n) = Sum_{k>=0} (x/(1-x))^(k^2).
Binomial transform of the characteristic function of squares A010052, with 0th term omitted. - Carl Najafi, Sep 09 2011
a(n) = Sum_{k >= 0} binomial(n-1,k^2-1). - Gus Wiseman, Jan 17 2019

a(0)=1 prepended by Alois P. Heinz, Jan 18 2019

A339445 Number of partitions of n into squares such that the number of parts is a square.

Original entry on oeis.org

1, 1, 0, 0, 2, 0, 0, 1, 0, 2, 1, 0, 2, 1, 0, 2, 3, 1, 2, 2, 2, 2, 2, 2, 3, 5, 2, 4, 6, 1, 4, 6, 3, 7, 6, 4, 10, 6, 4, 10, 9, 6, 11, 10, 8, 10, 10, 11, 14, 16, 11, 15, 19, 10, 17, 22, 13, 24, 23, 16, 28, 21, 18, 33, 30, 24, 33, 33, 29, 33, 37, 33, 43, 45, 35, 49

Offset: 0

Ilya Gutkovskiy, Dec 05 2020

nonn

			                                    [1 1 1]
                          [1 4]     [1 1 1]
a(23) = 2 because we have [9 9] and [4 4 9].

Robert Israel, Table of n, a(n) for n = 0..2500
Eric Weisstein's World of Mathematics, Square Number
Index entries for sequences related to partitions
Index entries for sequences related to sums of squares

Cf. A000290, A001156, A085755, A089333, A103198, A323522, A339444.

g:= proc(n, k, m)
  # number of partitions of n into k parts which are squares > m^2
   option remember; local r;
  if k = 0 then if n = 0 then return 1 else return 0 fi fi;
  if n < k*(m+1)^2 then return 0 fi;
  add(procname(n-r*(m+1)^2, k-r, m+1), r =max(0, ceil((k*(m+2)^2-n)/(2*m+3))) .. k)
end proc:
f:= proc(n) local k; add(g(n,k^2,0),k=1..floor(sqrt(n))) end proc:
f(0):= 1:
map(f, [$0..100]); # Robert Israel, Oct 26 2023

A357354 Number of partitions of n into distinct positive squares such that the number of parts is a square.

Original entry on oeis.org

1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 3, 1, 0, 2, 0, 0, 1, 1, 1

Offset: 0

Ilya Gutkovskiy, Sep 25 2022

			a(30) = 1 because we have [16,9,4,1].
a(78) = 3: [36,25,16,1], [49,16,9,4], [64,9,4,1].

Alois P. Heinz, Table of n, a(n) for n = 0..20000

Cf. A000290, A033461, A045450, A089333, A357352.

b:= proc(n, i, t) option remember; `if`(n=0,
     `if`(issqr(t), 1, 0), `if`(n>i*(i+1)*(2*i+1)/6, 0,
     `if`(i^2>n, 0, b(n-i^2, i-1, t+1))+b(n, i-1, t)))
    end:
a:= n-> b(n, isqrt(n), 0):
seq(a(n), n=0..100);  # Alois P. Heinz, Sep 25 2022

b[n_, i_, t_] := b[n, i, t] = If[n == 0,
   If[IntegerQ @ Sqrt[t], 1, 0], If[n > i*(i+1)*(2*i+1)/6, 0,
   If[i^2 > n, 0, b[n-i^2, i-1, t+1]] + b[n, i-1, t]]];
a[n_] := b[n, Floor @ Sqrt[n], 0];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Apr 24 2025, after Alois P. Heinz *)

A286141 Number of partitions of n into a squarefree number of parts.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 12, 16, 22, 30, 40, 53, 70, 92, 120, 154, 199, 254, 324, 409, 517, 648, 811, 1008, 1253, 1549, 1911, 2347, 2880, 3519, 4294, 5219, 6338, 7671, 9273, 11173, 13451, 16147, 19359, 23151, 27656, 32958, 39231, 46594, 55276, 65444, 77391, 91341, 107689, 126734

Offset: 0

Ilya Gutkovskiy, May 07 2017

nonn

Also number of partitions of n such that the largest part is a squarefree (A005117).

			a(6) = 9 because we have [6], [5, 1], [4, 2], [4, 1, 1], [3, 3], [3, 2, 1], [2, 2, 2], [2, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1] (partitions into a squarefree number of parts).
Also a(6) = 9 because we have [6], [5, 1], [3, 3], [3, 2, 1], [3, 1, 1, 1], [2, 2, 2], [2, 2, 1, 1], [2, 1, 1, 1, 1] and [1, 1, 1, 1, 1, 1] (partitions such that the largest part is a squarefree).

Index entries for related partition-counting sequences

Cf. A005117, A038499, A073576, A089333, A102848, A178927.

Join[{1}, Table[Length@Select[IntegerPartitions@n, SquareFreeQ@Length@# &], {n, 50}]]
nmax = 50; CoefficientList[Series[1 + Sum[MoebiusMu[i]^2 x^i/Product[1 - x^j, {j, 1, i}], {i, 1, nmax}], {x, 0, nmax}], x]

G.f.: 1 + Sum_{i>=1} x^A005117(i) / Product_{j=1..A005117(i)} (1 - x^j).

A334626 G.f.: Sum_{k>=0} x^(k^3) / Product_{j=1..k^3} (1 - x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 6, 8, 12, 16, 23, 30, 41, 53, 71, 90, 117, 147, 187, 231, 289, 354, 436, 528, 642, 770, 927, 1102, 1313, 1550, 1832, 2147, 2519, 2935, 3421, 3964, 4594, 5298, 6110, 7016, 8055, 9216, 10542, 12021, 13706, 15588, 17724, 20111

Offset: 0

Ilya Gutkovskiy, Dec 05 2020

nonn

Number of partitions of n such that the number of parts is a cube.

Also number of partitions of n such that the largest part is a cube.

			a(10) = 3 because we have [10], [3, 1, 1, 1, 1, 1, 1, 1] and [2, 2, 1, 1, 1, 1, 1, 1] (see the first comment) or [8, 2], [8, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1] (see the second comment).

Index entries for sequences related to partitions

Cf. A000578, A003108, A089333, A280351, A339235.

nmax = 53; CoefficientList[Series[Sum[x^(k^3)/Product[1 - x^j, {j, 1, k^3}], {k, 0, Floor[nmax^(1/3)] + 1}], {x, 0, nmax}], x]

A339235 G.f.: Sum_{k>=0} x^(k^4) / Product_{j=1..k^4} (1 - x^j).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 6, 8, 12, 16, 23, 31, 43, 57, 78, 102, 136, 177, 232, 297, 384, 487, 621, 781, 984, 1226, 1531, 1892, 2340, 2872, 3524, 4294, 5232, 6335, 7666, 9229, 11099, 13288, 15893, 18929, 22519, 26695, 31604, 37293

Offset: 0

Ilya Gutkovskiy, Dec 05 2020

nonn

Number of partitions of n such that the number of parts is a fourth power.

Also number of partitions of n such that the largest part is a fourth power.

Index entries for sequences related to partitions

Cf. A000583, A046042, A089333, A334626.

nmax = 57; CoefficientList[Series[Sum[x^(k^4)/Product[1 - x^j, {j, 1, k^4}], {k, 0, Floor[nmax^(1/4)] + 1}], {x, 0, nmax}], x]

a(18) = 3 because we have [18], [3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] and [2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] (see the first comment) or[16, 2], [16, 1, 1] and [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1] (see the second comment).

cp's OEIS Frontend

A103198 Number of compositions of n into a square number of parts.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A339445 Number of partitions of n into squares such that the number of parts is a square.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

A357354 Number of partitions of n into distinct positive squares such that the number of parts is a square.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A286141 Number of partitions of n into a squarefree number of parts.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A334626 G.f.: Sum_{k>=0} x^(k^3) / Product_{j=1..k^3} (1 - x^j).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A339235 G.f.: Sum_{k>=0} x^(k^4) / Product_{j=1..k^4} (1 - x^j).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula