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

0,7

Links

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

Formula

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A243148, A259254, A319503.

Sequence in context: A262811 A294749 A098359 * A226096 A071567 A304990

Adjacent sequences: A319432 A319433 A319434 * A319436 A319437 A319438

Keyword

nonn

Author

Alois P. Heinz, Sep 18 2018

Status

approved