A286653 -id:A286653 - cp's OEIS frontend

A000726 Number of partitions of n in which no parts are multiples of 3.

1, 1, 2, 2, 4, 5, 7, 9, 13, 16, 22, 27, 36, 44, 57, 70, 89, 108, 135, 163, 202, 243, 297, 355, 431, 513, 617, 731, 874, 1031, 1225, 1439, 1701, 1991, 2341, 2731, 3197, 3717, 4333, 5022, 5834, 6741, 7803, 8991, 10375, 11923, 13716, 15723, 18038, 20628, 23603

Offset: 0

N. J. A. Sloane

Case k=4, i=3 of Gordon Theorem.
Expansion of q^(-1/12)*eta(q^3)/eta(q) in powers of q. - Michael Somos, Apr 20 2004
Euler transform of period 3 sequence [1,1,0,...]. - Michael Somos, Apr 20 2004
Also the number of partitions with at most 2 parts of size 1 and all differences between parts at distance 3 are greater than 1. Example: a(6)=7 because we have [6],[5,1],[4,2],[4,1,1],[3,3],[3,2,1] and [2,2,2] (for example, [2,2,1,1] does not qualify because the difference between the first and the fourth parts is equal to 1). - Emeric Deutsch, Apr 18 2006
Also the number of partitions of n where no part appears more than twice. Example: a(6)=7 because we have [6],[5,1],[4,2],[4,1,1],[3,3],[3,2,1] and [2,2,1,1]. - Emeric Deutsch, Apr 18 2006
Also the number of partitions of n with least part either 1 or 2 and with differences of consecutive parts at most 2. Example: a(6)=7 because we have [4,2], [3,2,1], [3,1,1,1], [2,2,2], [2,2,1,1], [2,1,1,1,1] and [1,1,1,1,1,1]. - Emeric Deutsch, Apr 18 2006
Equals left border of triangle A174714. - Gary W. Adamson, Mar 27 2010
Triangle A113685 is equivalent to p(x) = p(x^2) * A000009(x); given A000041(x) = p(x). Triangle A176202 is equivalent to p(x) = p(x^3) * A000726(x). - Gary W. Adamson, Apr 11 2010
Convolution of A035382 and A035386. - Vaclav Kotesovec, Aug 23 2015
The number of partitions of n in which no parts are multiples of k equals the number of partitions of n where no part appears more than k-1 times. - Gregory L. Simay, Oct 15 2022

			There are a(6)=7 partitions of 6 into parts != 0 (mod 3):
[ 1]  [5,1],
[ 2]  [4,2],
[ 3]  [4,1,1],
[ 4]  [2,2,2],
[ 5]  [2,2,1,1],
[ 6]  [2,1,1,1,1], and
[ 7]  [1,1,1,1,1,1]
.
From _Joerg Arndt_, Dec 29 2012: (Start)
There are a(10)=22 partitions p(1)+p(2)+...+p(m)=10 such that p(k)!=p(k-2) (that is, no part appears more than twice):
[ 1]  [ 3 3 2 1 1 ]
[ 2]  [ 3 3 2 2 ]
[ 3]  [ 4 2 2 1 1 ]
[ 4]  [ 4 3 2 1 ]
[ 5]  [ 4 3 3 ]
[ 6]  [ 4 4 1 1 ]
[ 7]  [ 4 4 2 ]
[ 8]  [ 5 2 2 1 ]
[ 9]  [ 5 3 1 1 ]
[10]  [ 5 3 2 ]
[11]  [ 5 4 1 ]
[12]  [ 5 5 ]
[13]  [ 6 2 1 1 ]
[14]  [ 6 2 2 ]
[15]  [ 6 3 1 ]
[16]  [ 6 4 ]
[17]  [ 7 2 1 ]
[18]  [ 7 3 ]
[19]  [ 8 1 1 ]
[20]  [ 8 2 ]
[21]  [ 9 1 ]
[22]  [ 10 ]
(End)

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.
L. Carlitz, Generating functions and partition problems, pp. 144-169 of A. L. Whiteman, ed., Theory of Numbers, Proc. Sympos. Pure Math., 8 (1965). Amer. Math. Soc., see p. 145.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe and Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (terms 0..1000 from T. D. Noe)
George E. Andrews, Partition Identities for Two-Color Partitions, Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2021, Special Commemorative volume in honour of Srinivasa Ramanujan, 2021, 44, pp.74-80. hal-03498190. See p. 79.
Riccardo Aragona, Roberto Civino, and Norberto Gavioli, A modular idealizer chain and unrefinability of partitions with repeated parts, arXiv:2301.06347 [math.RA], 2023.
N. Chair, Partition identities from Partial Supersymmetry, arXiv:hep-th/0409011, 2004.
Edray Herber Goins and Talitha M. Washington, On the generalized climbing stairs problem, Ars Combin. 117 (2014), 183-190. MR3243840 (Reviewed), arXiv:0909.5459 [math.CO], 2009.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 15.
Eric Weisstein's World of Mathematics, Partition function b_k.
Wikipedia, Glaisher's Theorem.

Cf. A000009 (no multiples of 2), A001935 (no of 4), A035959 (no of 5), A219601 (no of 6), A035985, A001651, A003105, A035361, A035360.
Cf. A174714. - Gary W. Adamson, Mar 27 2010
Cf. A113685, A176202. - Gary W. Adamson, Apr 11 2010
Cf. A046913.
Column k=3 of A286653.
Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

a000726 n = p a001651_list n where
   p _  0 = 1
   p ks'@(k:ks) m | m < k     = 0
                  | otherwise = p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Aug 23 2011

g:=product(1+x^j+x^(2*j),j=1..60): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=0..50); # Emeric Deutsch, Apr 18 2006
# second Maple program:
with(numtheory):
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(
     `if`(irem(d, 3)=0, 0, d), d=divisors(j)), j=1..n)/n)
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Nov 17 2017

f[0] = 1; f[n_] := Coefficient[Expand@ Product[1 + x^k + x^(2k), {k, n}], x^n]; Table[f@n, {n, 0, 40}] (* Robert G. Wilson v, Nov 10 2006 *)
QP = QPochhammer; CoefficientList[QP[q^3]/QP[q] + O[q]^60, q] (* Jean-François Alcover, Nov 24 2015 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(3*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 02 2016 *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 3], 0, 2] ], {n, 0, 50}] (* Robert Price, Jul 28 2020 *)
Table[Count[IntegerPartitions[n],?(NoneTrue[Mod[#,3]==0&])],{n,0,50}] (* _Harvey P. Dale, Sep 06 2022 *)

a(n)=if(n<0,0,polcoeff(eta(x^3+x*O(x^n))/eta(x+x*O(x^n)),n))

lista(nn) = {q='q+O('q^nn); Vec(eta(q^3)/eta(q))} \\ Altug Alkan, Mar 20 2018

G.f.: 1/(Product_{k>=1} (1-x^(3*k-1))*(1-x^(3*k-2))) = Product_{k>=1} (1 + x^k + x^(2*k)) (where 1 + x + x^2 is the 3rd cyclotomic polynomial).
a(n) = A061197(n, n).
Given g.f. A(x) then B(x) = x*A(x^6)^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u,v,w) = +v^2 +v*w^2 -v*u^2 +3*u^2*w^2. - Michael Somos, May 28 2006
G.f.: P(x^3)/P(x) where P(x) = Product_{k>=1} (1 - x^k). - Joerg Arndt, Jun 21 2011
a(n) ~ 2*Pi * BesselI(1, sqrt((12*n + 1)/3)*Pi/3) / (3*sqrt(12*n + 1)) ~ exp(2*Pi*sqrt(n)/3) / (6*n^(3/4)) * (1 + (Pi/36 - 9/(16*Pi))/sqrt(n) + (Pi^2/2592 - 135/(512*Pi^2) - 5/64)/n). - Vaclav Kotesovec, Aug 23 2015, extended Jan 13 2017
a(n) = (1/n)*Sum_{k=1..n} A046913(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 21 2017
G.f.: exp(Sum_{k>=1} x^k*(1 + x^k)/(k*(1 - x^(3*k)))). - Ilya Gutkovskiy, Aug 15 2018

More terms from Olivier Gérard

A210485 Number T(n,k) of parts in all partitions of n in which no part occurs more than k times; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

Original entry on oeis.org

0, 0, 1, 0, 1, 3, 0, 3, 3, 6, 0, 3, 8, 8, 12, 0, 5, 11, 15, 15, 20, 0, 8, 17, 24, 29, 29, 35, 0, 10, 23, 36, 41, 47, 47, 54, 0, 13, 36, 50, 65, 71, 78, 78, 86, 0, 18, 48, 75, 91, 104, 111, 119, 119, 128, 0, 25, 69, 102, 132, 150, 165, 173, 182, 182, 192

Offset: 0

Alois P. Heinz, Jan 23 2013

T(n,k) is defined for n,k >= 0. The triangle contains terms with k <= n. T(n,k) = T(n,n) = A006128(n) for k >= n.

For fixed k > 0, T(n,k) ~ 3^(1/4) * log(k+1) * exp(Pi*sqrt(2*k*n/(3*(k+1)))) / (Pi * (8*k*(k+1)*n)^(1/4)). - Vaclav Kotesovec, Oct 18 2018

			T(6,2) = 17: [6], [5,1], [4,2], [3,3], [4,1,1], [3,2,1], [2,2,1,1].
Triangle T(n,k) begins:
  0;
  0,  1;
  0,  1,  3;
  0,  3,  3,  6;
  0,  3,  8,  8, 12;
  0,  5, 11, 15, 15, 20;
  0,  8, 17, 24, 29, 29, 35;
  0, 10, 23, 36, 41, 47, 47, 54;
  0, 13, 36, 50, 65, 71, 78, 78, 86;
  ...

Alois P. Heinz, Rows n = 0..140, flattened

Columns k=0-10 give: A000004, A015723, A185350, A117148, A320607, A320608, A320609, A320610, A320611, A320612, A320613.
Main diagonal gives A006128.
T(2n,n) gives A364245.
Cf. A213177, A286653.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->[l[1], l[2]+l[1]*j])(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
T:= (n, k)-> b(n, n, k)[2]:
seq(seq(T(n, k), k=0..n), n=0..12);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, Sum[b[n-i*j, i-1, k] /. l_List :> {l[[1]], l[[2]] + l[[1]]*j}, {j, 0, Min[n/i, k]}]]]; T[n_, k_] := b[n, n, k][[2]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

T(n,k) = Sum_{i=0..k} A213177(n,i).

A290307 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)/(1 + x^(k*j)).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 2, 1, 2, 1, 0, 1, 1, 1, 2, 2, 2, 2, 1, 0, 1, 1, 1, 2, 2, 2, 3, 3, 2, 0, 1, 1, 1, 2, 2, 3, 3, 3, 3, 2, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 3, 2, 0, 1, 1, 1, 2, 2, 3, 4, 4, 4, 5, 4, 2, 0, 1, 1, 1, 2, 2, 3, 4, 4, 5, 6, 6, 5, 3, 0

Offset: 0

Ilya Gutkovskiy, Jul 26 2017

A(n,k) is the number of partitions of n into distinct parts where no part is a multiple of k.

			Square array begins:
  1,  1,  1,  1,  1,  1, ...
  0,  1,  1,  1,  1,  1, ...
  0,  0,  1,  1,  1,  1, ...
  0,  1,  1,  2,  2,  2, ...
  0,  1,  1,  1,  2,  2, ...
  0,  1,  2,  2,  2,  3, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Index entries for sequences related to partitions

Columns k=1-10 give: A000007, A000700, A003105, A070048, A096938, A261770, A097793, A261771, A112193, A261772.

Cf. A286653.

Table[Function[k, SeriesCoefficient[Product[(1 + x^i)/(1 + x^(i k)), {i, Infinity}], {x, 0, n}]][j - n + 1], {j, 0, 13}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[-1, x]/QPochhammer[-1, x^k], {x, 0, n}]][j - n + 1], {j, 0, 13}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} (1 + x^j)/(1 + x^(k*j)).

For asymptotics of column k see comment from Vaclav Kotesovec in A261772.

A061198 Square table by antidiagonals where T(n,k) is number of partitions of k where no part appears more than n times; also partitions of k where no parts are multiples of (n+1).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 1, 1, 0, 2, 2, 2, 1, 1, 0, 3, 4, 3, 2, 1, 1, 0, 4, 5, 4, 3, 2, 1, 1, 0, 5, 7, 6, 5, 3, 2, 1, 1, 0, 6, 9, 9, 6, 5, 3, 2, 1, 1, 0, 8, 13, 12, 10, 7, 5, 3, 2, 1, 1, 0, 10, 16, 16, 13, 10, 7, 5, 3, 2, 1, 1, 0, 12, 22, 22, 19, 14, 11, 7, 5, 3, 2, 1, 1, 0, 15, 27, 29, 25, 20, 14, 11, 7, 5, 3, 2, 1, 1

Offset: 0

Henry Bottomley, Apr 20 2001

			Square table T(n,k) begins:
  1, 0, 0, 0, 0, 0,  0,  0,  0,  0,  0, ...
  1, 1, 1, 2, 2, 3,  4,  5,  6,  8, 10, ...
  1, 1, 2, 2, 4, 5,  7,  9, 13, 16, 22, ...
  1, 1, 2, 3, 4, 6,  9, 12, 16, 22, 29, ...
  1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 34, ...
  1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, ...
  1, 1, 2, 3, 5, 7, 11, 14, 21, 28, 39, ...
  1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, ...
  1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, ...
  1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, ...
  1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...

Rows include A000007, A000009, A000726, A035959.
Main diagonal is A000041.
A061199 is the same table but excluding cases where n>k.
Cf. A286653.

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1, k), j=0..min(n/i, k))))
    end:
A:= (n, k)-> b(k$2, n):
seq(seq(A(n, d-n), n=0..d), d=0..13);  # Alois P. Heinz, Jan 26 2023

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[b[n-i*j, i-1, k], {j, 0, Min[n/i, k]}]]];
A[n_, k_] := b[k, k, n];
Table[Table[A[n, d-n], {n, 0, d}], {d, 0, 13}] // Flatten (* Jean-François Alcover, Feb 11 2023, after Alois P. Heinz *)

G.f. for row n of table: Product_{j>=1} Sum_{k=0..n} x^(j*k) = Product_{j>=1} (1-x^((n+1)*j)) / (1-x^j). - Sean A. Irvine, Jan 26 2023

A341714 Coefficients in the expansion of Product_{m>=1} (1 - q^(13*m))/(1 - q^m).

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 100, 134, 174, 228, 292, 378, 479, 612, 770, 972, 1213, 1519, 1881, 2334, 2874, 3540, 4331, 5302, 6450, 7848, 9501, 11496, 13851, 16680, 20006, 23980, 28648, 34193, 40689, 48378, 57360, 67948, 80295, 94788, 111652, 131388, 154293

Offset: 0

N. J. A. Sloane, Feb 21 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000
J. J. Webb, Arithmetic of the 13-regular partition function modulo 3, Ramanujan Journal, 25 (2011), 49-56.

Column k=13 of A286653.

a(n) ~ exp(Pi*sqrt(2*n*(s-1)/(3*s))) * (s-1)^(1/4) / (2 * 6^(1/4) * s^(3/4) * n^(3/4)) * (1 + ((s-1)^(3/2)*Pi/(24*sqrt(6*s)) - 3*sqrt(6*s) / (16*Pi * sqrt(s-1))) / sqrt(n) + ((s-1)^3*Pi^2/(6912*s) - 45*s/(256*(s-1)*Pi^2) - 5*(s-1)/128) / n), set s=13. - Vaclav Kotesovec, Aug 01 2022

A304625 a(n) = [x^n] Product_{k>=1} ((1 - x^(n*k))/(1 - x^k))^n.

Original entry on oeis.org

1, 0, 3, 19, 101, 501, 2486, 12398, 62329, 315436, 1605330, 8207552, 42124368, 216903051, 1119974861, 5796944342, 30068145889, 156250892593, 813310723907, 4239676354631, 22130265931880, 115654632452514, 605081974091853, 3168828466966365, 16610409114771876, 87141919856550506

Offset: 0

Ilya Gutkovskiy, May 15 2018

nonn

Number of partitions of n into 2 or more parts of n kinds. - Ilya Gutkovskiy, May 16 2018

Index entries for sequences related to partitions

Cf. A000065, A008485, A022567, A093160, A270913, A285927, A285928, A286653, A296044, A296162, A296163, A304626.

Table[SeriesCoefficient[Product[((1 - x^(n k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 25}]
Table[SeriesCoefficient[Product[1/(1 - x^k)^n, {k, 1, n - 1}], {x, 0, n}], {n, 0, 25}]

a(n) ~ c * d^n / sqrt(n), where d = A270915 = 5.3527013334866426877724... and c = 0.268015212710733315686... - Vaclav Kotesovec, May 16 2018

A286656 Square array A(n,k), n>=0, k>=1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^j)/(1 - x^(k*j)).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -1, 0, 0, 1, -1, -1, -1, 0, 1, -1, -1, 1, 1, 0, 1, -1, -1, 0, -1, -1, 0, 1, -1, -1, 0, 1, 0, 1, 0, 1, -1, -1, 0, 0, 0, 2, -1, 0, 1, -1, -1, 0, 0, 2, -1, -1, 2, 0, 1, -1, -1, 0, 0, 1, -1, 1, -1, -2, 0, 1, -1, -1, 0, 0, 1, 1, 0, 2, 3, 2, 0, 1

Offset: 0

Seiichi Manyama, May 14 2017

			Square array begins:
   1,  1,  1,  1,  1, ...
   0, -1, -1, -1, -1, ...
   0,  0, -1, -1, -1, ...
   0, -1,  1,  0,  0, ...
   0,  1, -1,  1,  0, ...

Columns k=1-5 give: A000007, A081362, A137569, A082303, A145466.
Main diagonal gives A010815.
Cf. A286653.

G.f. of column k: Product_{j>=1} (1 - x^j)/(1 - x^(k*j)).

A302233 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^(k*j))/(1 + x^j).

Original entry on oeis.org

1, 1, 0, 1, -1, 0, 1, -1, 1, 0, 1, -1, 0, -2, 0, 1, -1, 0, 0, 2, 0, 1, -1, 0, -1, 0, -3, 0, 1, -1, 0, -1, 2, -1, 4, 0, 1, -1, 0, -1, 1, -2, 1, -5, 0, 1, -1, 0, -1, 1, 0, 1, -1, 6, 0, 1, -1, 0, -1, 1, -1, 0, -2, 1, -8, 0, 1, -1, 0, -1, 1, -1, 2, -1, 4, 0, 10, 0, 1, -1, 0, -1, 1, -1, 1, -2, 1, -4, 0, -12, 0

Offset: 0

Ilya Gutkovskiy, Apr 03 2018

			Square array begins:
1,  1,  1,  1,  1,  1,  ...
0, -1, -1, -1, -1, -1,  ...
0,  1,  0,  0,  0,  0,  ...
0, -2,  0, -1, -1, -1,  ...
0,  2,  0,  2,  1,  1,  ...
0, -3, -1, -2,  0, -1,  ...

Columns k=1-10 give: A000007, A081360, A109389, A261734, A133563, A261736, A113297, A261735, A261733, A145707.
Main diagonal gives A081362.
Cf. A286653, A286656, A290307.

Table[Function[k, SeriesCoefficient[Product[(1 + x^(k i))/(1 + x^i), {i, 1, n}], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[QPochhammer[-1, x^k]/QPochhammer[-1, x], {x, 0, n}]][j - n + 1], {j, 0, 12}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} (1 + x^(k*j))/(1 + x^j).

For asymptotics of column k see comment from Vaclav Kotesovec in A145707.

cp's OEIS Frontend

A000726 Number of partitions of n in which no parts are multiples of 3.

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A210485 Number T(n,k) of parts in all partitions of n in which no part occurs more than k times; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A290307 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^j)/(1 + x^(k*j)).

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A061198 Square table by antidiagonals where T(n,k) is number of partitions of k where no part appears more than n times; also partitions of k where no parts are multiples of (n+1).

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A341714 Coefficients in the expansion of Product_{m>=1} (1 - q^(13*m))/(1 - q^m).

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A304625 a(n) = [x^n] Product_{k>=1} ((1 - x^(n*k))/(1 - x^k))^n.

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A286656 Square array A(n,k), n>=0, k>=1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 - x^j)/(1 - x^(k*j)).

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A302233 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1 + x^(k*j))/(1 + x^j).

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