A087153 Number of partitions of n into nonsquares.

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, 178, 218, 257, 310, 366, 440, 515, 617, 722, 857, 1003, 1184, 1380, 1625, 1889, 2214, 2570, 3000, 3472, 4042, 4669, 5414, 6244, 7221, 8303, 9583, 10998, 12655, 14502

Offset

0,6

Comments

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.

Convolution of A276516 and A000041. - Vaclav Kotesovec, Dec 30 2016

From Gus Wiseman, Apr 02 2019: (Start)

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:

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

(32) (33) (43) (44) (54)

(42) (52) (53) (63)

(62) (72)

(332) (432)

while in the latter case they count the following:

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

(32) (33) (52) (53) (72)

(222) (322) (62) (333)

(332) (522)

(2222) (3222)

(End)

References

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

Links

T. D. Noe and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)

Daniel I. A. Cohen, PIE-sums: a combinatorial tool for partition theory. J. Combin. Theory Ser. A 31 (1981), no. 3, 223--236. MR0635367 (82m:10026). See Cor. 5. - N. J. A. Sloane, Mar 27 2012

James A. Sellers, Partitions Excluding Specific Polygonal Numbers As Parts, Journal of Integer Sequences, Vol. 7 (2004), Article 04.2.4.

Formula

G.f.: Product_{m>0} (1-x^(m^2))/(1-x^m). - Vladeta Jovovic, Aug 21 2003

a(n) = (1/n)*Sum_{k=1..n} (A000203(k)-A035316(k))*a(n-k), a(0)=1. - Vladeta Jovovic, Aug 21 2003

G.f.: Product_{i>=1} (Sum_{j=0..i-1} x^(i*j)). - Jon Perry, Jul 26 2004

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

Example

n=7: 2+5 = 2+2+3 = 7: a(7)=3;
n=8: 2+6 = 2+2+2+2 = 2+3+3 = 3+5 = 8: a(8)=5;
n=9: 2+7 = 2+2+5 = 2+2+2+3 = 3+3+3 = 3+6: a(9)=5.

Maple

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

Mathematica

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

Prog

(Haskell)
a087153 = p a000037_list where
   p _          0 = 1
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Apr 25 2013
(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

Crossrefs

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

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

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

Sequence in context: A120249 A058690 A290369 * A240176 A134408 A051032

Adjacent sequences: A087150 A087151 A087152 * A087154 A087155 A087156

Keyword

nonn

Author

Reinhard Zumkeller, Aug 21 2003

Extensions

Zero term added by Franklin T. Adams-Watters, Jan 25 2010

Status

approved