A264393 -id:A264393 - 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

A280204 G.f.: Product_{k>=1} (1+x^(k^2)) / (1-x^k).

Original entry on oeis.org

1, 2, 3, 5, 9, 14, 21, 31, 45, 65, 92, 127, 175, 239, 322, 430, 572, 753, 985, 1281, 1657, 2131, 2727, 3471, 4401, 5558, 6988, 8751, 10924, 13588, 16846, 20819, 25653, 31518, 38621, 47195, 57530, 69958, 84869, 102723, 124070, 149532, 179852, 215894, 258668

Offset: 0

Vaclav Kotesovec, Dec 28 2016

nonn

Convolution of A033461 and A000041.

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

Cf. A000041, A033461, A052335, A078434, A087153, A117144, A264393, A280224, A280225, A369520, A369570.

nmax=50; CoefficientList[Series[Product[(1+x^(k^2))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ exp(Pi*sqrt(2*n/3) + 2^(-5/4)*3^(1/4)*(sqrt(2)-1)*Zeta(3/2)*n^(1/4) - 3*(sqrt(2)-1)^2*Zeta(3/2)^2/(64*Pi)) / (2^(5/2)*sqrt(3)*n).

A369520 Expansion of Product_{k>=1} 1/((1 - x^(k^2))*(1 - x^k)).

Original entry on oeis.org

1, 2, 4, 7, 13, 21, 34, 52, 80, 119, 175, 251, 359, 504, 702, 965, 1320, 1785, 2401, 3200, 4245, 5589, 7324, 9535, 12364, 15944, 20478, 26175, 33338, 42279, 53438, 67283, 84454, 105642, 131764, 163826, 203149, 251185, 309799, 381079, 467666, 572520, 699342, 852314

Offset: 0

Vaclav Kotesovec, Jan 25 2024

nonn

Convolution of A001156 and A000041.

a(n) is the number of pairs (Q(k), P(n-k)), 0<=k<=n, where Q(k) is a partition of k into squares and P(n-k) is a partition of n-k.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (250000 terms)

Cf. A000041, A001156, A087153, A264393, A280204, A369519, A369579.

nmax=50; CoefficientList[Series[Product[1/(1-x^(k^2))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

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)) / (2^(13/4) * 3^(3/4) * sqrt(Pi) * n^(5/4)).

A280278 G.f.: Product_{k>=1} (1 + x^(k^3)) / (1 - x^k).

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 18, 26, 38, 54, 75, 103, 141, 190, 254, 337, 444, 580, 754, 973, 1250, 1597, 2030, 2568, 3237, 4061, 5076, 6322, 7847, 9705, 11968, 14711, 18033, 22043, 26873, 32677, 39642, 47972, 57924, 69787, 83904, 100667, 120547, 144072, 171876, 204677

Offset: 0

Vaclav Kotesovec, Dec 30 2016

nonn

Convolution of A279329 and A000041.

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

Cf. A000041, A264393, A279329, A280204, A369571, A369579.

nmax=60; CoefficientList[Series[Product[(1+x^(k^3))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

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

A369519 Expansion of Product_{k>=1} 1/((1 - x^(k^2))*(1 - x^(k^3))).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 16, 21, 26, 31, 38, 46, 54, 62, 74, 88, 103, 118, 137, 158, 180, 202, 230, 263, 298, 335, 378, 426, 476, 528, 589, 658, 732, 810, 900, 998, 1101, 1208, 1330, 1465, 1608, 1760, 1930, 2116, 2310, 2513, 2738, 2985, 3246, 3521, 3826, 4156

Offset: 0

Vaclav Kotesovec, Jan 25 2024

nonn

Convolution of A001156 and A003108.

a(n) is the number of pairs (Q(k), P(n-k)), 0<=k<=n, where Q(k) is a partition of k into squares and P(n-k) is a partition of n-k into cubes.

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

Cf. A001156, A003108.

Cf. A087153, A131799, A264393, A369520, A369579.

nmax=100; CoefficientList[Series[Product[1/(1-x^(k^2))/(1-x^(k^3)), {k, 1, nmax^(1/2)}], {x, 0, nmax}], x]

a(n) ~ zeta(3/2) * exp(3 * Pi^(1/3) * zeta(3/2)^(2/3) * n^(1/3) / 2^(4/3) + 2^(4/9) * Gamma(1/3) * zeta(4/3) * n^(2/9) / (3 * Pi^(1/9) * zeta(3/2)^(2/9)) - 4*2^(2/9) * Gamma(1/3)^2 * zeta(4/3)^2 * n^(1/9) / (243 * Pi^(5/9) * zeta(3/2)^(10/9)) + 16*Gamma(1/3)^3 * zeta(4/3)^3 / (6561 * Pi * zeta(3/2)^2)) / (16 * sqrt(6) * Pi^(5/2) * n^(3/2)) * (1 + (13*2^(7/9) * Gamma(1/3) * zeta(4/3) / (81 * Pi^(4/9) * zeta(3/2)^(8/9)) + 832*2^(7/9) * Gamma(1/3)^4 * zeta(4/3)^4 / (1594323 * Pi^(13/9) * zeta(3/2)^(26/9))) / n^(1/9) + (692224 * 2^(5/9) * Gamma(1/3)^8 * zeta(4/3)^8 / (2541865828329 * Pi^(26/9) * zeta(3/2)^(52/9)) - 128 * 2^(5/9) * Gamma(1/3)^5 * zeta(4/3)^5 / (4782969 * Pi^(17/9) * zeta(3/2)^(34/9)) + 65*2^(5/9) * Gamma(1/3)^2 * zeta(4/3)^2 / (2187*Pi^(8/9) * zeta(3/2)^(16/9)))/n^(2/9)).

A264391 Triangle read by rows: T(n,k) is the number of partitions of n having k perfect cube parts (0<=k<=n).

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 4, 2, 2, 1, 1, 0, 1, 4, 4, 2, 2, 1, 1, 0, 1, 6, 5, 4, 2, 2, 1, 1, 0, 1, 8, 6, 5, 4, 2, 2, 1, 1, 0, 1, 11, 9, 6, 5, 4, 2, 2, 1, 1, 0, 1, 13, 12, 9, 6, 5, 4, 2, 2, 1, 1, 0, 1, 19, 15, 12, 9, 6, 5, 4, 2, 2, 1, 1, 0, 1

Offset: 0

Emeric Deutsch, Nov 13 2015

Sum of entries in row n = A000041(n) = number of partitions of n.
T(n,0) = A264393(n).
Sum_{k=0..n}k*T(n,k) = A264392(n) = total number of perfect cube parts in all partitions of n.

			T(7,1) = 4 because we have [6,1],[4,2,1],[3,3,1], and [2,2,2,1] (the partitions of 7 that have 1 perfect cube part).
Triangle starts:
  1;
  0, 1;
  1, 0, 1;
  1, 1, 0, 1;
  2, 1, 1, 0, 1;
  2, 2, 1, 1, 0, 1;

Alois P. Heinz, Rows n = 0..200, flattened

Cf. A000041, A264392, A264393.

h := proc (i) options operator, arrow: i^3 end proc: g := product((1-x^h(i))/((1-x^i)*(1-t*x^h(i))), i = 1 .. 80): gser := simplify(series(g, x = 0, 30)): for n from 0 to 18 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 18 do seq(coeff(P[n], t, j), j = 0 .. n) end do; # yields sequence in triangular form
# second Maple program:
q:= proc(n) option remember; `if`(n=iroot(n, 3)^3, 1, 0) end:
b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+x^q(i)*b(n-i, min(i, n-i)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..20);  # Alois P. Heinz, Nov 14 2020

cnt[P_List] := Count[P, p_ /; IntegerQ[p^(1/3)]];
cnts[n_] := cnts[n] = cnt /@ IntegerPartitions[n];
T[n_, k_] := Count[cnts[n], k];
Table[T[n, k], {n, 0, 18}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 14 2020 *)

G.f.: G(t,x) = Product_{i>=1} (1-x^h(i))/((1-x^i)*(1-t*x^h(i))), where h(i) = i^3.

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

A280204 G.f.: Product_{k>=1} (1+x^(k^2)) / (1-x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A369520 Expansion of Product_{k>=1} 1/((1 - x^(k^2))*(1 - x^k)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A280278 G.f.: Product_{k>=1} (1 + x^(k^3)) / (1 - x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A369519 Expansion of Product_{k>=1} 1/((1 - x^(k^2))*(1 - x^(k^3))).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A264391 Triangle read by rows: T(n,k) is the number of partitions of n having k perfect cube parts (0<=k<=n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula