OFFSET
0,26
COMMENTS
"WEIGH" transform of squares A000290.
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
The Heinz numbers of integer partitions into distinct pairs are given by A324587. - Gus Wiseman, Mar 09 2019
From Gus Wiseman, Mar 09 2019: (Start)
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:
1: (1)
4: (22)
5: (221)
9: (333)
10: (3331)
13: (33322)
14: (333221)
16: (4444)
17: (44441)
20: (444422)
21: (4444221)
25: (55555)
25: (4444333)
26: (555551)
26: (44443331)
29: (5555522)
29: (444433322)
30: (55555221)
30: (4444333221)
The case where the distinct parts are taken in decreasing order is A324572, with Heinz numbers given by A324571.
(End)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
M. Brack, M. V. N. Murthy, and J. Bartel, Application of semiclassical methods to number theory, University of Regensburg (Germany, 2020).
Martin Klazar, What is an answer? — remarks, results and problems on PIO formulas in combinatorial enumeration, part I, arXiv:1808.08449 [math.CO], 2018.
Vaclav Kotesovec, Graph - The asymptotic ratio.
M. V. N. Murthy, Matthias Brack, Rajat K. Bhaduri, and Johann Bartel, Semi-classical analysis of distinct square partitions, arXiv:1808.05146 [cond-mat.stat-mech], 2018.
FORMULA
G.f.: Product_{n>=1} ( 1+x^(n^2) ).
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
See Murthy, Brack, Bhaduri, Bartel (2018) for a more complete asymptotic expansion. - N. J. A. Sloane, Aug 17 2018
EXAMPLE
a(50)=3 because we have [1,4,9,36], [1,49], and [9,16,25]. - Emeric Deutsch, Jan 26 2016
From Gus Wiseman, Mar 09 2019: (Start)
The first 30 terms count the following integer partitions:
1: (1)
4: (4)
5: (4,1)
9: (9)
10: (9,1)
13: (9,4)
14: (9,4,1)
16: (16)
17: (16,1)
20: (16,4)
21: (16,4,1)
25: (25)
25: (16,9)
26: (25,1)
26: (16,9,1)
29: (25,4)
29: (16,9,4)
30: (25,4,1)
30: (16,9,4,1)
(End)
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1) +`if`(i^2>n, 0, b(n-i^2, i-1))))
end:
a:= n-> b(n, isqrt(n)):
seq(a(n), n=0..100); # Alois P. Heinz, May 14 2014
MATHEMATICA
nn=10; CoefficientList[Series[Product[(1+x^(k*k)), {k, nn}], {x, 0, nn*nn}], x] (* T. D. Noe, Jul 24 2006 *)
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 *)
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 *)
Table[Length[Select[IntegerPartitions[n], Reverse[Union[#]]==Length/@Split[#]&]], {n, 30}] (* Gus Wiseman, Mar 09 2019 *)
PROG
(PARI) a(n)=polcoeff(prod(k=1, sqrt(n), 1+x^k^2), n)
(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
(Python)
from functools import cache
from sympy.core.power import isqrt
@cache
def b(n, i):
# Code after Alois P. Heinz
if n == 0: return 1
if i == 0: return 0
i2 = i*i
return b(n, i-1) + (0 if i2 > n else b(n - i2, i-1))
a = lambda n: b(n, isqrt(n))
print([a(n) for n in range(1, 101)]) # Darío Clavijo, Nov 30 2023
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Michael Somos
STATUS
approved