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A033461 Number of partitions of n into distinct squares. 75
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,26

COMMENTS

"WEIGH" transform of squares A000290.

a(n) = 0 for n in {A001422}, a(n) > 0 for n in {A003995}. - Alois P. Heinz, May 14 2014

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

LINKS

T. D. Noe and Alois P. Heinz, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)

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, 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

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 *)

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

CROSSREFS

Cf. A003995, A001422, A242434 (the same for compositions), A078434, A279329.

Sequence in context: A113406 A151851 A321447 * A143432 A137677 A015818

Adjacent sequences:  A033458 A033459 A033460 * A033462 A033463 A033464

KEYWORD

nonn,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Michael Somos

STATUS

approved

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Last modified November 20 02:57 EST 2018. Contains 317371 sequences. (Running on oeis4.)