login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A087153 Number of partitions of n into nonsquares. 27

%I #49 Apr 02 2019 08:04:58

%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

%C From _Gus Wiseman_, Apr 02 2019: (Start)

%C The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). The Heinz numbers of the integer partitions described in Perry and Jovovic's comment are given by A325128, while the Heinz numbers of the integer partitions described in the name are given by A325129. In the former case, the first 10 terms count the following integer partitions:

%C () (2) (3) (4) (5) (6) (7) (8) (9)

%C (32) (33) (43) (44) (54)

%C (42) (52) (53) (63)

%C (62) (72)

%C (332) (432)

%C while in the latter case they count the following:

%C () (2) (3) (22) (5) (6) (7) (8) (63)

%C (32) (33) (52) (53) (72)

%C (222) (322) (62) (333)

%C (332) (522)

%C (2222) (3222)

%C (End)

%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 (Haskell)

%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.

%Y Cf. A033461, A114639, A117144, A276429, A324572, A324588, A325128, A325129.

%K nonn

%O 0,6

%A _Reinhard Zumkeller_, Aug 21 2003

%E Zero term added by _Franklin T. Adams-Watters_, Jan 25 2010

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 11:44 EDT 2024. Contains 371241 sequences. (Running on oeis4.)