A089299 -id:A089299 - cp's OEIS frontend

A008763 Expansion of g.f.: x^4/((1-x)(1-x^2)^2(1-x^3)).

0, 0, 0, 0, 1, 1, 3, 4, 7, 9, 14, 17, 24, 29, 38, 45, 57, 66, 81, 93, 111, 126, 148, 166, 192, 214, 244, 270, 305, 335, 375, 410, 455, 495, 546, 591, 648, 699, 762, 819, 889, 952, 1029, 1099, 1183, 1260, 1352, 1436, 1536, 1628, 1736, 1836, 1953, 2061, 2187, 2304, 2439

Offset: 0

N. J. A. Sloane, Simon Plouffe, Stephen P. Humphries

Number of 2 X 2 square partitions of n.
1/((1-x^2)*(1-x^4)^2*(1-x^6)) is the Molien series for 4-dimensional representation of a certain group of order 192 [Nebe, Rains, Sloane, Chap. 7].
Number of ways of writing n as n = p+q+r+s so that p >= q, p >= r, q >= s, r >= s with p, q, r, s >= 1. That is, we can partition n as
pq
rs
with p >= q, p >= r, q >= s, r >= s.
The coefficient of s(2n) in s(n,n) * s(n,n) * s(n,n) * s(n,n) is a(n+4), where s(n) is the Schur function corresponding to the trivial representation, s(n,n) is a Schur function corresponding to the two row partition and * represents the inner or Kronecker product of symmetric functions. - Mike Zabrocki, Dec 22 2005
Let F() be the Fibonacci sequence A000045. Let f([x, y, z, w]) = F(x) * F(y) * F(z) * F(w). Let N([x, y, z, w]) = x^2 + y^2 + z^2 + w^2. Let Q(k) = set of all ordered quadruples of integers [x, y, z, w] such that 1 <= x <= y <= z <= w and N([x, y, z, w]) = k. Let P(n) = set of all unordered triples {q1, q2, q3} of elements of some Q(k) such that max(w1, w2, w3) = n and f(q1) + f(q2) = f(q3). Then a(n-1) is the number of elements of P(n). - Michael Somos, Jan 21 2015
Number of partitions of 2n+2 into 4 parts with alternating parity from smallest to largest (or vice versa). - Wesley Ivan Hurt, Jan 19 2021

			a(7) = 4:
41 32 31 22
11 11 21 21
G.f. = x^4 + x^5 + 3*x^6 + 4*x^7 + 7*x^8 + 9*x^9 + 14*x^10 + 17*x^11 + ...
a(5-1) = 1 because P(5) has only one triple {[1,1,1,5], [2,2,2,4], [1,3,3,3]} of elements from Q(28) where f([1,1,1,5]) = 5, f([2,2,2,4]) = 3, f([1,3,3,3]) = 8, and 5 + 3 = 8. - _Michael Somos_, Jan 21 2015
a(6-1) = 1 because P(6) has only one triple {[1,1,2,6], [2,2,3,5], [1,3,4,4]} of elements from Q(42) where f([1,1,2,6]) = 8, f([2,2,3,5]) = 10, f([1,3,4,4]) = 18 and 8 + 10 = 18. - _Michael Somos_, Jan 21 2015
a(7-1) = 3 because P(7) has three triples. The triple {[1,1,1,7], [2,4,4,4], [3,3,3,5]} from Q(52) where f([1,1,1,7]) = 13, f([2,4,4,4]) = 27, f([3,3,3,5]) = 40 and 13 + 27 = 40. The triple {[1,2,2,7], [2,3,3,6], [1,4,4,5]} from Q(58) where f([1,2,2,7]) = 13, f([2,3,3,6]) = 32, f([1,4,4,5]) = 45 and 13 + 32 = 45. The triple {[1,1,3,7], [2,2,4,6], [1,3,5,5]} from Q(60) where f([1,1,3,7]) = 26, f([2,2,4,6]) = 24, f([1,3,5,5]) = 50 and 26 + 24 = 50. - _Michael Somos_, Jan 21 2015

G. E. Andrews, MacMahon's Partition Analysis II: Fundamental Theorems, Annals Combinatorics, 4 (2000), 327-338.
G. E. Andrews, P. Paule and A. Riese, MacMahon's Partition Analysis VIII: Plane partition diamonds, Advances Applied Math., 27 (2001), 231-242 (Cor. 2.1, n=1).
S. P. Humphries, Braid groups, infinite Lie algebras of Cartan type and rings of invariants, Topology and its Applications, 95 (3) (1999) pp. 173-205.

T. D. Noe, Table of n, a(n) for n = 0..1000
Nesrine Benyahia-Tani, Zahra Yahi, and Sadek Bouroubi, Ordered and non-ordered non-congruent convex quadrilaterals inscribed in a regular n-gon, Rostocker Math. Kolloq. 68, 71-79 (2013), Theorem 5.
W. Duke, On codes and Siegel modular forms, Int. Math. Res. Notes 1993, No. 5, Theorem 2. [MR1219862 (94d:11029)]
Michele Graffeo, Sergej Monavari, Riccardo Moschetti, and Andrea T. Ricolfi, Enumeration of partitions via socle reduction, arXiv:2501.10267 [math.CO], 2025. See p. 40.
S. P. Humphries, Home page
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 450
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 232
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Michael Somos, In the Elliptic Realm
Index entries for linear recurrences with constant coefficients, signature (1,2,-1,-2,-1,2,1,-1).

See A266769 for a version without the four leading zeros.
Cf. A001993, A070557, A070558, A070559, A089299, A001970, A089292, A026810, A001400.
First differences of A097701.
Cf. A082424, A082437.

a:=[0,0,0,0,1,1,3,4];; for n in [9..60] do a[n]:=a[n-1]+2*a[n-2]-a[n-3]-2*a[n-4]-a[n-5]+2*a[n-6]+a[n-7]-a[n-8]; od; a; # G. C. Greubel, Sep 10 2019

K:=Rationals(); M:=MatrixAlgebra(K,4); q1:=DiagonalMatrix(M,[1,-1,1,-1]); p1:=DiagonalMatrix(M,[1,1,-1,-1]); q2:=DiagonalMatrix(M,[1,1,1,-1]); h:=M![1,1,1,1, 1,1,-1,-1, 1,-1,1,-1, 1,-1,-1,1]/2; H:=MatrixGroup<4,K|q1,q2,h,p1>; MolienSeries(H);

R:=PowerSeriesRing(Integers(), 60); [0,0,0,0] cat Coefficients(R!( x^4/((1-x)*(1-x^2)^2*(1-x^3)) )); // G. C. Greubel, Sep 10 2019

a:= n-> (Matrix(8, (i,j)-> if (i=j-1) then 1 elif j=1 then [1,2,-1,-2,-1,2,1,-1][i] else 0 fi)^n)[1,5]: seq(a(n), n=0..60); # Alois P. Heinz, Jul 31 2008

CoefficientList[Series[x^4/((1-x)*(1-x^2)^2*(1-x^3)), {x,0,60}], x] (* Jean-François Alcover, Mar 30 2011 *)
LinearRecurrence[{1,2,-1,-2,-1,2,1,-1},{0,0,0,0,1,1,3,4},60] (* Harvey P. Dale, Mar 04 2012 *)
a[ n_]:= Quotient[9(n+1)(-1)^n +2n^3 -9n +65, 144]; (* Michael Somos, Jan 21 2015 *)
a[ n_]:= Sign[n] SeriesCoefficient[ x^4/((1-x)(1-x^2)^2(1-x^3)), {x, 0, Abs@n}]; (* Michael Somos, Jan 21 2015 *)

{a(n) = (9*(n+1)*(-1)^n + 2*n^3 - 9*n + 65) \ 144}; /* Michael Somos, Jan 21 2015 */

a(n)=([0,1,0,0,0,0,0,0; 0,0,1,0,0,0,0,0; 0,0,0,1,0,0,0,0; 0,0,0,0,1,0,0,0; 0,0,0,0,0,1,0,0; 0,0,0,0,0,0,1,0; 0,0,0,0,0,0,0,1; -1,1,2,-1,-2,-1,2,1]^n*[0;0;0;0;1;1;3;4])[1,1] \\ Charles R Greathouse IV, Feb 06 2017

def AA008763_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P(x^4/((1-x)*(1-x^2)^2*(1-x^3))).list()
AA008763_list(60) # G. C. Greubel, Sep 10 2019

Let f4(n) = number of partitions n = p+q+r+s into exactly 4 parts, with p >= q >= r >= s >= 1 (see A026810, A001400) and let g4(n) be the number with q > r (so that g4(n) = f4(n-2)). Then a(n) = f4(n) + g4(n).
a(n) = (1/144)*( 2*n^3 + 9*n*((-1)^n - 1) - 16*((n is 2 mod 3) - (n is 1 mod 3)) ).
a(n) = (1/72)*(n+3)*(n+2)*(n+1)-(1/12)*(n+2)*(n+1)+(5/144)*(n+1)+(1/16)*(n+1)*(-1)^n+(1/16)*(-1)^(n+1)+(7/144)+(2*sqrt(3)/27)*sin(2*Pi*n/3). - Richard Choulet, Nov 27 2008
a(n) = a(n-1) + 2*a(n-2) - a(n-3) - 2*a(n-4) - a(n-5) + 2*a(n-6) + a(n-7) - a(n-8), n>7. - Harvey P. Dale, Mar 04 2012
a(n) = floor((9*(n+1)*(-1)^n + 2*n^3 - 9*n + 65)/144). - Tani Akinari, Nov 06 2012
a(n+1) - a(n) = A008731(n-3). - R. J. Mathar, Aug 06 2013
a(n) = -a(-n) for all n in Z. - Michael Somos, Jan 21 2015
Euler transform of length 3 sequence [1, 2, 1]. - Michael Somos, Jun 26 2017

Entry revised Dec 25 2003

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

Original entry on oeis.org

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

A306319 Number of length-rectangular twice-partitions of n.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 26, 35, 60, 82, 131, 177, 286, 376, 582, 793, 1202, 1610, 2450, 3274, 4906, 6665, 9770, 13274, 19690, 26506, 38596, 53006, 76432, 104189, 150844, 205282, 294304, 404146, 573140, 786169, 1119457, 1527554, 2155953, 2965567, 4163955, 5701816

Offset: 0

Gus Wiseman, Feb 07 2019

nonn

A twice partition of n is a sequence of integer partitions, one of each part in an integer partition of n. It is length-rectangular if all parts have the same number of parts.

			The a(5) = 14 length-rectangular twice-partitions:
  [5] [4 1] [3 2] [3 1 1] [2 2 1] [2 1 1 1] [1 1 1 1 1]
.
  [4] [3] [2 1]
  [1] [2] [1 1]
.
  [3] [2]
  [1] [2]
  [1] [1]
.
  [2]
  [1]
  [1]
  [1]
.
  [1]
  [1]
  [1]
  [1]
  [1]

Dominates A319066 (rectangular partitions of partitions), which dominates A323429 (rectangular plane partitions).

Cf. A000219, A001970, A063834 (twice-partitions), A089299, A271619, A279787 (sum-rectangular twice-partitions), A305551, A306017, A306318 (square case), A323531.

Table[Length[Join@@Table[Select[Tuples[IntegerPartitions/@ptn],SameQ@@Length/@#&],{ptn,IntegerPartitions[n]}]],{n,20}]

A323432 Number of semistandard rectangular plane partitions of n.

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 15, 20, 30, 42, 59, 79, 112, 146, 199, 264, 350, 455, 603, 774, 1010, 1297, 1668, 2124, 2724, 3441, 4372, 5513, 6955, 8718, 10960, 13670, 17091, 21264, 26454, 32786, 40667, 50215, 62048, 76435, 94126

Offset: 0

Gus Wiseman, Jan 16 2019

nonn

Number of ways to fill a (not necessarily square) matrix with the parts of an integer partition of n so that the rows are weakly decreasing and the columns are strictly decreasing.

			The a(6) = 15 matrices:
  [6] [51] [42] [411] [33] [321] [3111] [222] [2211] [21111] [111111]
.
  [5] [4] [22]
  [1] [2] [11]
.
  [3]
  [2]
  [1]

Cf. A000219, A003293, A047966, A089299, A101509, A114736, A117433, A299968, A319066.

Cf. A323301, A323307, A323429, A323430, A323431, A323435, A323436, A323438.

Table[Sum[Length[Select[Union[Tuples[IntegerPartitions[#,{k}]&/@ptn]],And@@(OrderedQ[#,Greater]&/@Transpose[#])&]],{ptn,IntegerPartitions[n]},{k,Min[ptn]}],{n,30}]

A323431 Number of strict rectangular plane partitions of n.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 7, 9, 11, 15, 21, 25, 33, 41, 53, 65, 81, 97, 121, 143, 173, 215, 255, 305, 367, 441, 527, 637, 751, 899, 1067, 1269, 1491, 1775, 2071, 2439, 2875, 3357, 3911, 4577, 5309, 6177, 7171, 8305, 9609, 11151

Offset: 0

Gus Wiseman, Jan 16 2019

nonn

Number of ways to fill a (not necessarily square) matrix with the parts of a strict integer partition of n so that the rows and columns are strictly decreasing.

			The a(10) = 21 matrices:
  [10] [9 1] [8 2] [7 3] [7 2 1] [6 4] [6 3 1] [5 4 1] [5 3 2] [4 3 2 1]
.
  [9] [8] [7] [6] [4 2] [4 3]
  [1] [2] [3] [4] [3 1] [2 1]
.
  [7] [6] [5] [5]
  [2] [3] [4] [3]
  [1] [1] [1] [2]
.
  [4]
  [3]
  [2]
  [1]

Cf. A000219, A003293, A047966, A089299 A101509, A114736, A117433, A299968, A319066.

Cf. A323301, A323307, A323429, A323430, A323432, A323435, A323436, A323438.

Table[Sum[Length[Select[Union[Sort/@Tuples[IntegerPartitions[#,{k}]&/@ptn]],UnsameQ@@Join@@#&&And@@OrderedQ/@Transpose[#]&]],{ptn,IntegerPartitions[n]},{k,Min[ptn]}],{n,30}]

A323519 a(n) is the number of ways to fill a square matrix with the multiset of prime factors of n, if the number of prime factors (counted with multiplicity) is a perfect square, and a(n) = 0 otherwise.

Original entry on oeis.org

1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 4, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 6, 1, 0, 0, 4, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 4, 0, 4, 0, 0, 1, 12, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 12, 0, 0

Offset: 1

Gus Wiseman, Jan 17 2019

nonn

			The a(60) = 12 matrices:
  [2 2] [2 2] [2 3] [2 3] [2 5] [2 5] [3 2] [3 2] [3 5] [5 2] [5 2] [5 3]
  [3 5] [5 3] [2 5] [5 2] [2 3] [3 2] [2 5] [5 2] [2 2] [2 3] [3 2] [2 2]

Positions of 0's are A323521.
Positions of 1's are A323520.
Cf. A000290, A008480, A026478, A056239, A089299, A103198, A112798, A120732, A323433, A323525, A323531.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[If[IntegerQ[Sqrt[PrimeOmega[n]]],Length[Permutations[primeMS[n]]],0],{n,100}]

If A001222(n) is a perfect square, then a(n) = A008480(n). Otherwise, a(n) = 0.

A323529 Number of strict square plane partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 5, 7, 11, 13, 19, 23, 31, 37, 47, 55, 69, 79, 95, 109, 129, 145, 169, 189, 217, 241, 273, 301, 339, 371, 413, 451, 499, 541, 595, 643, 703, 757, 823, 925, 999, 1107, 1229, 1387, 1559, 1807, 2071, 2453, 2893, 3451, 4109, 5011

Offset: 0

Gus Wiseman, Jan 17 2019

nonn

			The a(12) = 5 strict square plane partitions:
  [12]
.
  [1 2] [1 2] [1 3] [1 4]
  [3 6] [4 5] [2 6] [2 5]
The a(15) = 13 strict square plane partitions:
  [15]
.
  [7 5] [8 4] [9 3] [6 5] [7 4] [9 2] [6 4] [7 3] [8 2] [6 3] [6 3] [7 2]
  [2 1] [2 1] [2 1] [3 1] [3 1] [3 1] [3 2] [4 1] [4 1] [4 2] [5 1] [5 1]

Alois P. Heinz, Table of n, a(n) for n = 0..7500

Cf. A000219, A008289, A039622, A047966, A089299, A101509, A319066, A323429, A323434, A323522, A323530.

h:= proc(n) h(n):= (n^2)!*mul(k!/(n+k)!, k=0..n-1) end:
b:= proc(n, i, t) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, `if`(issqr(t), h(isqrt(t)), 0),
         b(n, i-1, t) +b(n-i, min(n-i, i-1), t+1)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..70);  # Alois P. Heinz, Jan 24 2019

Table[Sum[Length[Select[Union[Sort/@Tuples[Reverse/@IntegerPartitions[#,{Length[ptn]}]&/@ptn]],UnsameQ@@Join@@#&&And@@OrderedQ/@Transpose[#]&]],{ptn,IntegerPartitions[n]}],{n,30}]
(* Second program: *)
h[n_] := (n^2)! Product[k!/(k+n)!, {k, 0, n-1}];
b[n_, i_, t_] := b[n, i, t] = If[n > i(i+1)/2, 0, If[n == 0, If[IntegerQ[ Sqrt[t]], h[Sqrt[t]], 0], b[n-i, Min[n-i, i-1], t+1] + b[n, i-1, t]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 70] (* Jean-François Alcover, May 19 2021, after Alois P. Heinz *)

a(n) = Sum_{j>=0} A039622(j) * A008289(n,j^2). - Alois P. Heinz, Jan 24 2019

More terms from Alois P. Heinz, Jan 24 2019

A323522 Number of ways to fill a square matrix with the parts of a strict integer partition of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25, 25, 49, 73, 121, 145, 217, 265, 361, 433, 553, 649, 817, 937, 1129, 1297, 1537, 1729, 2017, 2257, 2593, 2881, 3265, 3601, 4057, 4441, 4945, 5401, 5977, 6481, 7129, 7705, 8425, 9073, 9865, 373465, 374353, 738025, 1101865, 1828513

Offset: 0

Gus Wiseman, Jan 17 2019

nonn

			The a(10) = 25 matrices:
  [10]
.
  [4 3] [4 3] [4 2] [4 2] [4 1] [4 1] [3 4] [3 4]
  [2 1] [1 2] [3 1] [1 3] [3 2] [2 3] [2 1] [1 2]
.
  [3 2] [3 2] [3 1] [3 1] [2 4] [2 4] [2 3] [2 3]
  [4 1] [1 4] [4 2] [2 4] [3 1] [1 3] [4 1] [1 4]
.
  [2 1] [2 1] [1 4] [1 4] [1 3] [1 3] [1 2] [1 2]
  [4 3] [3 4] [3 2] [2 3] [4 2] [2 4] [4 3] [3 4]

Alois P. Heinz, Table of n, a(n) for n = 0..7000

Cf. A000009, A089299, A103198 (non-strict case), A120732, A323431, A323434, A323519, A323523, A323525, A323529.

b:= proc(n, i) b(n, i):= `if`(n=0, [1], `if`(i<1, [], zip((x, y)
      -> x+y, b(n, i-1), `if`(i>n, [], [0, b(n-i, i-1)[]]), 0)))
    end:
a:= n-> (l-> add(l[i^2+1]*(i^2)!, i=0..floor(sqrt(nops(l)-1))))(b(n$2)):
seq(a(n), n=0..50);  # Alois P. Heinz, Jan 17 2019

Table[Sum[(k^2)!*Length[Select[IntegerPartitions[n,{k^2}],UnsameQ@@#&]],{k,n}],{n,20}]
(* Second program: *)
q[n_, k_] := q[n, k] = If[n < k || k < 1, 0,
     If[n == 1, 1, q[n-k, k] + q[n-k, k-1]]];
a[n_] := If[n == 0, 1, Sum[(k^2)! q[n, k^2], {k, 0, n}]];
a /@ Range[0, 50] (* Jean-François Alcover, May 20 2021 *)

a(n) = Sum_{k >= 0} (k^2)! * Q(n, k^2) where Q = A008289.

A306318 Number of square twice-partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 5, 10, 12, 19, 24, 39, 49, 73, 104, 151, 212, 317, 443, 638, 936, 1296, 1841, 2635, 3641, 5069, 7176, 9884, 13614, 19113, 26162, 36603, 50405, 70153, 96176, 135388, 184753, 257882, 353587, 494653, 671992, 934905, 1272195, 1762979, 2389255

Offset: 0

Gus Wiseman, Feb 07 2019

nonn

A twice partition of n is a sequence of integer partitions, one of each part in an integer partition of n. It is square if the number of parts is equal to the number of parts in each part.

			The a(10) = 19 square twice-partitions:
  ((ten))  ((32)(32))  ((211)(111)(111))
           ((32)(41))
           ((33)(22))
           ((33)(31))
           ((41)(32))
           ((41)(41))
           ((42)(22))
           ((42)(31))
           ((43)(21))
           ((44)(11))
           ((51)(22))
           ((51)(31))
           ((52)(21))
           ((53)(11))
           ((61)(21))
           ((62)(11))
           ((71)(11))

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000219, A001970, A063834 (twice-partitions), A089299 (square plane partitions), A279787, A305551, A306017, A306319 (rectangular twice-partitions), A319066, A323429, A323531 (square partitions of partitions).

Table[Sum[Length[Union@@(Tuples[IntegerPartitions[#,{k}]&/@#]&/@IntegerPartitions[n,{k}])],{k,0,Sqrt[n]}],{n,0,20}]

cp's OEIS Frontend

A008763 Expansion of g.f.: x^4/((1-x)*(1-x^2)^2*(1-x^3)).

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

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A306319 Number of length-rectangular twice-partitions of n.

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A323432 Number of semistandard rectangular plane partitions of n.

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A323431 Number of strict rectangular plane partitions of n.

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A323519 a(n) is the number of ways to fill a square matrix with the multiset of prime factors of n, if the number of prime factors (counted with multiplicity) is a perfect square, and a(n) = 0 otherwise.

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A323529 Number of strict square plane partitions of n.

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A008763 Expansion of g.f.: x^4/((1-x)(1-x^2)^2(1-x^3)).