A319435 Number of partitions of n^2 into exactly n nonzero squares.

1, 1, 0, 1, 1, 1, 4, 4, 9, 16, 24, 52, 83, 152, 305, 515, 959, 1773, 3105, 5724, 10255, 18056, 32584, 58082, 101719, 179306, 317610, 552730, 962134, 1683435, 2899064, 4995588, 8638919, 14746755, 25196684, 43082429, 72959433, 123554195, 209017908, 351164162

Offset: 0

Author

Alois P. Heinz, Sep 18 2018

Keywords

• nonn

Examples

			a(0) = 1: the empty partition.
a(1) = 1: 1.
a(2) = 0: there is no partition of 4 into exactly 2 nonzero squares.
a(3) = 1: 441.
a(4) = 1: 4444.
a(5) = 1: 94444.
a(6) = 4: (25)44111, (16)(16)1111, (16)44444, 999441.
a(7) = 4: (25)(16)41111, (25)444444, (16)(16)44441, (16)999411.
a(8) = 9: (49)9111111, (36)(16)441111, (36)4444444, (25)(25)911111, (25)(16)944411, (25)9999111, (16)(16)(16)94111, (16)9999444, 99999991.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A243148, A259254, A319503.

Programs

Maple

h:= proc(n) option remember; `if`(n<1, 0,
      `if`(issqr(n), n, h(n-1)))
    end:
b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1 or
      t<1, 0, b(n, h(i-1), t)+b(n-i, h(min(n-i, i)), t-1)))
    end:
a:= n-> (s-> b(s$2, n)-`if`(n=0, 0, b(s$2, n-1)))(n^2):
seq(a(n), n=0..40);

Mathematica

h[n_] := h[n] = If[n < 1, 0, If[Sqrt[n] // IntegerQ, n, h[n - 1]]];
b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, If[i < 1 || t < 1, 0, b[n, h[i - 1], t] + b[n - i, h[Min[n - i, i]], t - 1]]];
a[n_] := Function[s, b[s, s, n] - If[n == 0, 0, b[s, s, n - 1]]][n^2];
a /@ Range[0, 40] (* Jean-François Alcover, Nov 06 2020, after Alois P. Heinz *)

SageMath

# uses[GeneralizedEulerTransform(n, a) from A338585], slow.
def A319435List(n): return GeneralizedEulerTransform(n, lambda n: n^2)
print(A319435List(10)) # Peter Luschny, Nov 12 2020

Formula

a(n) = A243148(n^2,n).