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%I #87 Dec 03 2023 09:12:43
%S 1,1,0,0,1,1,0,0,0,1,1,0,0,1,1,0,1,1,0,0,1,1,0,0,0,2,2,0,0,2,2,0,0,0,
%T 1,1,1,1,1,1,1,2,1,0,0,2,2,0,0,2,3,1,1,2,2,1,1,1,1,1,0,2,3,1,1,4,3,0,
%U 1,2,2,1,0,1,4,3,0,2,4,2,1,3,2,1,2,3,3,2,1,3,6,3,0,2,5,3,0,1,3,3,3,4
%N Number of partitions of n into distinct squares.
%C "WEIGH" transform of squares A000290.
%C a(n) = 0 for n in {A001422}, a(n) > 0 for n in {A003995}. - _Alois P. Heinz_, May 14 2014
%C Number of partitions of n in which each part i has multiplicity i. Example: a(50)=3 because we have [1,2,2,3,3,3,6,6,6,6,6,6], [1,7,7,7,7,7,7,7], and [3,3,3,4,4,4,4,5,5,5,5,5]. - _Emeric Deutsch_, Jan 26 2016
%C The Heinz numbers of integer partitions into distinct pairs are given by A324587. - _Gus Wiseman_, Mar 09 2019
%C From _Gus Wiseman_, Mar 09 2019: (Start)
%C Equivalent to _Emeric Deutsch_'s comment, a(n) is the number of integer partitions of n where the multiplicities (where if x < y the multiplicity of x is counted prior to the multiplicity of y) are equal to the distinct parts in increasing order. The Heinz numbers of these partitions are given by A109298. For example, the first 30 terms count the following integer partitions:
%C 1: (1)
%C 4: (22)
%C 5: (221)
%C 9: (333)
%C 10: (3331)
%C 13: (33322)
%C 14: (333221)
%C 16: (4444)
%C 17: (44441)
%C 20: (444422)
%C 21: (4444221)
%C 25: (55555)
%C 25: (4444333)
%C 26: (555551)
%C 26: (44443331)
%C 29: (5555522)
%C 29: (444433322)
%C 30: (55555221)
%C 30: (4444333221)
%C The case where the distinct parts are taken in decreasing order is A324572, with Heinz numbers given by A324571.
%C (End)
%H Alois P. Heinz, <a href="/A033461/b033461.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from T. D. Noe)
%H M. Brack, M. V. N. Murthy, and J. Bartel, <a href="https://homepages.uni-regensburg.de/~brm04014/notes/F2Kaz4.pdf">Application of semiclassical methods to number theory</a>, University of Regensburg (Germany, 2020).
%H Martin Klazar, <a href="http://arxiv.org/abs/1808.08449">What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I</a>, arXiv:1808.08449 [math.CO], 2018.
%H Vaclav Kotesovec, <a href="/A033461/a033461.jpg">Graph - The asymptotic ratio</a>.
%H M. V. N. Murthy, Matthias Brack, Rajat K. Bhaduri, and Johann Bartel, <a href="https://arxiv.org/abs/1808.05146">Semi-classical analysis of distinct square partitions</a>, arXiv:1808.05146 [cond-mat.stat-mech], 2018.
%F G.f.: Product_{n>=1} ( 1+x^(n^2) ).
%F a(n) ~ exp(3 * 2^(-5/3) * Pi^(1/3) * ((sqrt(2)-1)*zeta(3/2))^(2/3) * n^(1/3)) * ((sqrt(2)-1)*zeta(3/2))^(1/3) / (2^(4/3) * sqrt(3) * Pi^(1/3) * n^(5/6)), where zeta(3/2) = A078434. - _Vaclav Kotesovec_, Dec 09 2016
%F See Murthy, Brack, Bhaduri, Bartel (2018) for a more complete asymptotic expansion. - _N. J. A. Sloane_, Aug 17 2018
%e a(50)=3 because we have [1,4,9,36], [1,49], and [9,16,25]. - _Emeric Deutsch_, Jan 26 2016
%e From _Gus Wiseman_, Mar 09 2019: (Start)
%e The first 30 terms count the following integer partitions:
%e 1: (1)
%e 4: (4)
%e 5: (4,1)
%e 9: (9)
%e 10: (9,1)
%e 13: (9,4)
%e 14: (9,4,1)
%e 16: (16)
%e 17: (16,1)
%e 20: (16,4)
%e 21: (16,4,1)
%e 25: (25)
%e 25: (16,9)
%e 26: (25,1)
%e 26: (16,9,1)
%e 29: (25,4)
%e 29: (16,9,4)
%e 30: (25,4,1)
%e 30: (16,9,4,1)
%e (End)
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p b(n, i-1) +`if`(i^2>n, 0, b(n-i^2, i-1))))
%p end:
%p a:= n-> b(n, isqrt(n)):
%p seq(a(n), n=0..100); # _Alois P. Heinz_, May 14 2014
%t nn=10; CoefficientList[Series[Product[(1+x^(k*k)), {k,nn}], {x,0,nn*nn}], x] (* _T. D. Noe_, Jul 24 2006 *)
%t b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i^2 > n, 0, b[n - i^2, i-1]]]]; a[n_] := b[n, Floor[Sqrt[n]]]; Table[a[n], {n, 0, 100}] (* _Jean-François Alcover_, Sep 21 2015, after _Alois P. Heinz_ *)
%t nmax = 20; poly = ConstantArray[0, nmax^2 + 1]; poly[[1]] = 1; poly[[2]] = 1; Do[Do[poly[[j + 1]] += poly[[j - k^2 + 1]], {j, nmax^2, k^2, -1}];, {k, 2, nmax}]; poly (* _Vaclav Kotesovec_, Dec 09 2016 *)
%t Table[Length[Select[IntegerPartitions[n],Reverse[Union[#]]==Length/@Split[#]&]],{n,30}] (* _Gus Wiseman_, Mar 09 2019 *)
%o (PARI) a(n)=polcoeff(prod(k=1,sqrt(n),1+x^k^2), n)
%o (PARI) first(n)=Vec(prod(k=1,sqrtint(n),1+'x^k^2,O('x^(n+1))+1)) \\ _Charles R Greathouse IV_, Sep 03 2015
%o (Python)
%o from functools import cache
%o from sympy.core.power import isqrt
%o @cache
%o def b(n,i):
%o # Code after _Alois P. Heinz_
%o if n == 0: return 1
%o if i == 0: return 0
%o i2 = i*i
%o return b(n, i-1) + (0 if i2 > n else b(n - i2, i-1))
%o a = lambda n: b(n, isqrt(n))
%o print([a(n) for n in range(1, 101)]) # _Darío Clavijo_, Nov 30 2023
%Y Cf. A001422, A003995, A078434, A242434 (the same for compositions), A279329.
%Y Cf. A001156 (non-strict case), A001462, A005117, A052335, A078135, A109298, A114638, A117144, A324571, A324572, A324587, A324588.
%Y Row sums of A341040.
%K nonn,nice
%O 0,26
%A _N. J. A. Sloane_
%E More terms from _Michael Somos_
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