The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS"). Other ways to donate

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 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-François 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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

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