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 A087153 Number of partitions of n into nonsquares. 16

%I

%S 1,0,1,1,1,2,3,3,5,5,8,9,13,15,20,24,30,37,47,55,71,83,103,123,151,

%T 178,218,257,310,366,440,515,617,722,857,1003,1184,1380,1625,1889,

%U 2214,2570,3000,3472,4042,4669,5414,6244,7221,8303,9583,10998,12655,14502

%N Number of partitions of n into nonsquares.

%C Also, number of partitions of n where there are fewer than k parts equal to k for all k. - _Jon Perry_ and _Vladeta Jovovic_, Aug 04 2004. E.g. a(8)=5 because we have 8=6+2=5+3=4+4=3+3+2.

%C Convolution of A276516 and A000041. - _Vaclav Kotesovec_, Dec 30 2016

%D G. E. Andrews, K. Eriksson, Integer Partitions, Cambridge Univ. Press, 2004. See page 48.

%H T. D. Noe and Vaclav Kotesovec, <a href="/A087153/b087153.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from T. D. Noe)

%H Daniel I. A. Cohen, <a href="http://dx.doi.org/10.1016/0097-3165(81)90057-1">PIE-sums: a combinatorial tool for partition theory</a>. J. Combin. Theory Ser. A 31 (1981), no. 3, 223--236. MR0635367 (82m:10026). See Cor. 5. - _N. J. A. Sloane_, Mar 27 2012

%H James A. Sellers, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL7/Sellers/sellers58.html">Partitions Excluding Specific Polygonal Numbers As Parts</a>, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.

%F G.f.: Product_{m>0} (1-x^(m^2))/(1-x^m). - _Vladeta Jovovic_, Aug 21 2003

%F a(n) = (1/n)*Sum_{k=1..n} (A000203(k)-A035316(k))*a(n-k), a(0)=1. - _Vladeta Jovovic_, Aug 21 2003

%F G.f.: product(i>=1, sum(j=0..i-1, x^(i*j) )). - _Jon Perry_, Jul 26 2004

%F a(n) ~ exp(Pi*sqrt(2*n/3) - 3^(1/4) * Zeta(3/2) * n^(1/4) / 2^(3/4) - 3*Zeta(3/2)^2/(32*Pi)) * sqrt(Pi) / (2^(3/4) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Dec 30 2016

%e n=7: 2+5 = 2+2+3 = 7: a(7)=3;

%e n=8: 2+6 = 2+2+2+2 = 2+3+3 = 3+5 = 8: a(8)=5;

%e n=9: 2+7 = 2+2+5 = 2+2+2+3 = 3+3+3 = 3+6: a(9)=5.

%p g:=product((1-x^(i^2))/(1-x^i),i=1..70):gser:=series(g,x=0,60):seq(coeff(gser,x^n),n=1..53); # _Emeric Deutsch_, Feb 09 2006

%t nn=54; CoefficientList[ Series[ Product[ Sum[x^(i*j), {j, 0, i - 1}], {i, 1, nn}], {x, 0, nn}], x] (* _Robert G. Wilson v_, Aug 05 2004 *)

%t nmax = 100; CoefficientList[Series[Product[(1 - x^(k^2))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Dec 29 2016 *)

%o a087153 = p a000037_list where

%o p _ 0 = 1

%o p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m

%o -- _Reinhard Zumkeller_, Apr 25 2013

%o (PARI) first(n)=my(x='x+O('x^(n+1))); Vec(prod(m=1,sqrtint(n), (1-x^m^2)/(1-x^m))*prod(m=sqrtint(n)+1,n,1/(1-x^m))) \\ _Charles R Greathouse IV_, Aug 28 2016

%Y Cf. A087154, A001156, A000009, A000037, A052335 (<=k parts of k).

%Y Cf. A115584, A172151, A225044, A264393, A276516.

%K nonn

%O 0,6

%A _Reinhard Zumkeller_, Aug 21 2003