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A319435 Number of partitions of n^2 into exactly n nonzero squares. 6

%I

%S 1,1,0,1,1,1,4,4,9,16,24,52,83,152,305,515,959,1773,3105,5724,10255,

%T 18056,32584,58082,101719,179306,317610,552730,962134,1683435,2899064,

%U 4995588,8638919,14746755,25196684,43082429,72959433,123554195,209017908,351164162

%N Number of partitions of n^2 into exactly n nonzero squares.

%H Alois P. Heinz, <a href="/A319435/b319435.txt">Table of n, a(n) for n = 0..200</a>

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

%e a(0) = 1: the empty partition.

%e a(1) = 1: 1.

%e a(2) = 0: there is no partition of 4 into exactly 2 nonzero squares.

%e a(3) = 1: 441.

%e a(4) = 1: 4444.

%e a(5) = 1: 94444.

%e a(6) = 4: (25)44111, (16)(16)1111, (16)44444, 999441.

%e a(7) = 4: (25)(16)41111, (25)444444, (16)(16)44441, (16)999411.

%e 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.

%p h:= proc(n) option remember; `if`(n<1, 0,

%p `if`(issqr(n), n, h(n-1)))

%p end:

%p b:= proc(n, i, t) option remember; `if`(n=0, 1, `if`(i<1 or

%p t<1, 0, b(n, h(i-1), t)+b(n-i, h(min(n-i, i)), t-1)))

%p end:

%p a:= n-> (s-> b(s$2, n)-`if`(n=0, 0, b(s$2, n-1)))(n^2):

%p seq(a(n), n=0..40);

%t h[n_] := h[n] = If[n < 1, 0, If[Sqrt[n] // IntegerQ, n, h[n - 1]]];

%t 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]]];

%t a[n_] := Function[s, b[s, s, n] - If[n == 0, 0, b[s, s, n - 1]]][n^2];

%t a /@ Range[0, 40] (* _Jean-Fran├žois Alcover_, Nov 06 2020, after _Alois P. Heinz_ *)

%o (SageMath) # uses[GeneralizedEulerTransform(n, a) from A338585], slow.

%o def A319435List(n): return GeneralizedEulerTransform(n, lambda n: n^2)

%o print(A319435List(10)) # _Peter Luschny_, Nov 12 2020

%Y Cf. A243148, A259254, A319503.

%K nonn

%O 0,7

%A _Alois P. Heinz_, Sep 18 2018

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Last modified December 4 15:01 EST 2020. Contains 338925 sequences. (Running on oeis4.)