A087154 -id:A087154 - cp's OEIS frontend

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

Reinhard Zumkeller, Aug 21 2003

nonn

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)

			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.

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

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.

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.

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

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

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 *)

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

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

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

A280276 G.f.: Product_{k>=1} (1 + x^k) / (1 - x^(k^2)).

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 17, 24, 33, 46, 62, 82, 108, 141, 182, 233, 297, 375, 472, 590, 733, 907, 1117, 1369, 1671, 2034, 2465, 2978, 3586, 4304, 5152, 6149, 7319, 8689, 10293, 12162, 14340, 16871, 19806, 23207, 27139, 31678, 36909, 42932, 49851, 57794, 66897

Offset: 0

Vaclav Kotesovec, Dec 30 2016

nonn

Convolution of A000009 and A001156.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000009, A001156, A087154, A103265, A280277, A369570, A369574.

nmax=80; CoefficientList[Series[Product[(1+x^k)/(1-x^(k^2)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(n/3) + 3^(1/4) * Zeta(3/2) * n^(1/4) / sqrt(2) - 3*Zeta(3/2)^2 / (16*Pi)) / (8*sqrt(6*Pi)*n).

A225045 Number of partitions of n into distinct non-triangular numbers, cf. A014132.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 3, 5, 5, 5, 7, 7, 10, 10, 13, 13, 16, 18, 21, 25, 27, 32, 33, 41, 44, 53, 57, 65, 73, 81, 93, 102, 118, 128, 145, 159, 181, 200, 224, 246, 275, 304, 337, 375, 413, 460, 503, 559, 614, 679, 749, 821, 907, 991, 1096, 1197, 1319, 1442, 1582, 1733, 1893, 2076, 2265, 2482, 2702, 2956, 3220

Offset: 0

Reinhard Zumkeller, Apr 25 2013

nonn

			a(10) = #{8+2} = 1;
a(11) = #{11, 9+2, 7+4, 5+4+2} = 4;
a(12) = #{12, 8+4, 7+5} = 3;
a(13) = #{13, 11+2, 9+4, 8+5, 7+4+2} = 5.

Alois P. Heinz and Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Alois P. Heinz)

Cf. A000009, A024940, A225044, A087154.

a225045 = p a014132_list where
   p _      0 = 1
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m

b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0,
      `if`(n=0, 1, add(b(n-i*j, i-1), j=0..min(n/i,
      `if`(issqr(8*i+1), 0, 1)))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..80);  # Alois P. Heinz, Apr 01 2014

b[n_, i_] := b[n, i] = If[n > i*(i+1)/2, 0, If[n==0, 1, Sum[b[n-i*j, i-1], {j, 0, Min[n/i, If[IntegerQ[Sqrt[8*i+1]], 0, 1]]}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jan 15 2016, after Alois P. Heinz *)

N=66; q='q+O('q^N); Vec( prod(n=1,N, 1 + q^n) / prod(n=1,N, 1 + q^(n*(n+1)/2)) ) \\ Joerg Arndt, Apr 01 2014

G.f.: prod(n>=1, 1 + q^n ) / prod(n>=1, 1 + q^(n*(n+1)/2) ). [Joerg Arndt, Apr 01 2014]

a(n) ~ exp(Pi*sqrt(n/3) - 3^(1/4) * Zeta(3/2) * n^(1/4) / (2+sqrt(2)) - 3*(3-2*sqrt(2)) * Zeta(3/2)^2 / (16*Pi)) / (2*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jan 02 2017

A280264 G.f.: Product_{k>=1} (1+x^k) / (1+x^(k^3)).

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 2, 5, 4, 6, 7, 8, 10, 12, 15, 16, 21, 23, 28, 33, 38, 44, 51, 60, 68, 79, 91, 103, 120, 136, 156, 177, 202, 230, 260, 296, 333, 378, 425, 480, 540, 606, 682, 764, 857, 958, 1073, 1197, 1337, 1492, 1660, 1849, 2057, 2285, 2537, 2816

Offset: 0

Vaclav Kotesovec, Dec 30 2016

nonn

Number of partitions of n into distinct noncubes (A007412). - Ilya Gutkovskiy, Dec 31 2016

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000009, A279329, A087154.

nmax=100; CoefficientList[Series[Product[(1+x^k)/(1+x^(k^3)), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(n/3) - (2^(1/3) - 1) * Gamma(1/3) * Zeta(4/3) * n^(1/6) / (3^(5/6) * Pi^(1/3))) / (2^(3/2) * 3^(1/4) * n^(3/4)).

cp's OEIS Frontend

A087153 Number of partitions of n into nonsquares.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

Extensions

A280276 G.f.: Product_{k>=1} (1 + x^k) / (1 - x^(k^2)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A225045 Number of partitions of n into distinct non-triangular numbers, cf. A014132.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Formula

A280264 G.f.: Product_{k>=1} (1+x^k) / (1+x^(k^3)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula