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 A033461 Number of partitions of n into distinct squares. 95
 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 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 T. D. Noe and 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, 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 CROSSREFS Cf. A001422, A003995, A078434, A242434 (the same for compositions), A279329. Cf. A001156 (non-strict case), A001462, A005117, A052335, A078135, A109298, A114638, A117144, A324571, A324572, A324587, A324588. Row sums of A341040. Sequence in context: A151851 A321447 A341698 * A143432 A137677 A015818 Adjacent sequences:  A033458 A033459 A033460 * A033462 A033463 A033464 KEYWORD nonn,nice AUTHOR EXTENSIONS More terms from Michael Somos STATUS approved

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Last modified April 17 12:40 EDT 2021. Contains 343063 sequences. (Running on oeis4.)