A293482 -id:A293482 - cp's OEIS frontend

A085311 Number of distinct 8th powers modulo n.

1, 2, 2, 2, 2, 4, 4, 2, 4, 4, 6, 4, 4, 8, 4, 2, 3, 8, 10, 4, 8, 12, 12, 4, 6, 8, 10, 8, 8, 8, 16, 2, 12, 6, 8, 8, 10, 20, 8, 4, 6, 16, 22, 12, 8, 24, 24, 4, 22, 12, 6, 8, 14, 20, 12, 8, 20, 16, 30, 8, 16, 32, 16, 3, 8, 24, 34, 6, 24, 16, 36, 8, 10, 20, 12, 20, 24, 16, 40, 4, 28, 12, 42, 16, 6

Offset: 1

Labos Elemer, Jun 27 2003

This sequence is multiplicative. - Leon P Smith, Apr 16 2005

T. D. Noe, Table of n, a(n) for n = 1..1000
S. Li, On the number of elements with maximal order in the multiplicative group modulo n, Acta Arithm. 86 (2) (1998) 113, see proof of theorem 2.1
R. J. Mathar, Size of the set of residues of integer powers of fixed exponent, (2017).

Cf. A000224[k=2], A046530[k=3], A052273[k=4], A052274[k=5], A052275[k=6], A085310[k=7], A085312[k=9], A085313[k=10], A085314[k=11], A228849[k=12], A055653.

A085311 := proc(m)
    {seq( modp(b^8,m),b=0..m-1) };
    nops(%) ;
end proc:
seq(A085311(m),m=1..100) ; # R. J. Mathar, Sep 22 2017

a[n_] := Table[PowerMod[i, 8, n], {i, 0, n - 1}] // Union // Length;
Array[a, 100] (* Jean-François Alcover, Mar 24 2020 *)

a(n)=my(f=factor(n)); prod(i=1, #f[, 1], my(k=f[i, 1]^f[i, 2]); #vecsort(vector(k, i, i^8%k), , 8)) \\ Charles R Greathouse IV, Sep 05 2013

A319099 Number of solutions to x^5 == 1 (mod n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 5, 1, 1, 1, 1, 1, 5, 1, 5, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 5, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 5, 5, 1, 1, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 5, 1, 5, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1

Offset: 1

Jianing Song, Sep 10 2018

All terms are powers of 5. Those n such that a(n) > 1 are in A066500.

			Solutions to x^5 == 1 (mod 11): x == 1, 3, 4, 5, 9 (mod 11).
Solutions to x^5 == 1 (mod 25): x == 1, 6, 11, 16, 21 (mod 25) (x == 1 (mod 5)).
Solutions to x^5 == 1 (mod 31): x == 1, 2, 4, 8, 16 (mod 31).

Jianing Song, Table of n, a(n) for n = 1..10000

Number of solutions to x^k == 1 (mod n): A060594 (k=2), A060839 (k=3), A073103 (k=4), this sequence (k=5), A319100 (k=6), A319101 (k=7), A247257 (k=8).
Cf. A030430, A066500, A293482, A000010.
Mobius transform gives A307380.

f[p_, e_] := If[Mod[p, 5] == 1, 5, 1]; f[5, 1] = 1; f[5, e_] := 5; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 10 2023 *)

a(n)=my(Z=znstar(n)[2]); prod(i=1,#Z,gcd(5,Z[i]));

Multiplicative with a(5) = 1, a(5^e) = 5 if e >= 2; for other primes p, a(p^e) = 5 if p == 1 (mod 5), a(p^e) = 1 otherwise.
If the multiplicative group of integers modulo n is isomorphic to C_{k_1} x C_{k_2} x ... x C_{k_m}, where k_i divides k_j for i < j; then a(n) = Product_{i=1..m} gcd(5, k_i).
a(n) = A000010(n)/A293482(n). - Jianing Song, Nov 10 2019

A288341 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)...(1-x^6)).

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 44, 64, 90, 125, 169, 227, 298, 388, 498, 634, 797, 996, 1231, 1513, 1844, 2235, 2689, 3221, 3833, 4542, 5353, 6284, 7341, 8547, 9907, 11447, 13176, 15121, 17293, 19725, 22427, 25436, 28767, 32459, 36529, 41023, 45958, 51385, 57327

Offset: 0

Seiichi Manyama, Jun 08 2017

Number of partitions of at most n into at most 6 parts.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Richard J. Mathar, Size of the Set of Residues of Integer Powers of Fixed Exponent, 2017.
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1, 0, -1, 1, -2, 2, 1, 0, 0, 0, -1, -2, 2, -1, 1, 0, 1, 0, -2, 1).

Number of partitions of at most n into at most k parts: A002621 (k=4), A002622 (k=5), this sequence (k=6), A288342 (k=7), A288343 (k=8), A288344 (k=9), A288345 (k=10).

Cf. A288253. Column 6 of A092905. A001402 (first differences).

x='x+O('x^99); Vec(1/((1-x)*prod(i=1, 6, (1-x^i)))) \\ Altug Alkan, Mar 28 2018

A092905 Triangle, read by rows, such that the partial sums of the n-th row form the n-th diagonal, for n>=0, where each row begins with 1.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 4, 2, 1, 1, 5, 6, 4, 2, 1, 1, 6, 9, 7, 4, 2, 1, 1, 7, 12, 11, 7, 4, 2, 1, 1, 8, 16, 16, 12, 7, 4, 2, 1, 1, 9, 20, 23, 18, 12, 7, 4, 2, 1, 1, 10, 25, 31, 27, 19, 12, 7, 4, 2, 1, 1, 11, 30, 41, 38, 29, 19, 12, 7, 4, 2, 1, 1, 12, 36, 53, 53, 42, 30, 19, 12, 7, 4, 2, 1

Offset: 0

Paul D. Hanna, Mar 12 2004

Row sums form A000070, which is the partial sums of the partition numbers (A000041). Rows read backwards converge to the row sums (A000070).
From Alford Arnold, Feb 07 2010: (Start)
The table can also be generated by summing sequences embedded within Table A008284
For example,
1 1 1 1 ... yields 1 2 3 4 ...
1 1 2 2 3 3 ... yields 1 2 4 6 9 12 ...
1 1 2 3 4 5 7 ... yields 1 2 4 7 11 16 ...
(End)
T(n,k) is also count of all 'replacable' cells in the (Ferrers plots of) the partitions on n in exactly k parts. [Wouter Meeussen, Sep 16 2010]
From Wolfdieter Lang, Dec 03 2012: (Start)
The triangle entry T(n,k) is obtained from triangle A072233 by summing the entries of column k up to n (see the partial sum type o.g.f. given by Vladeta Jovovic in the formula section).
  Therefore, the  o.g.f. for the sequence in column k is x^k/((1-x)* product(1-x^j,j=1..k)).
The triangle with entry a(n,m) = T(n-1,m-1), n >= 1, m = 1, ..., n, is obtained from the partition array A103921 when in row n all entries belonging to part number m are summed (a conjecture). (End)

			The fourth row (n=3) is {1,3,2,1} and the fourth diagonal is the partial sums of the fourth row: {1,4,6,7,7,7,7,7,...}.
The triangle T(n,k) begins:
n\k 0  1  2  3  4  5  6  7  8  9 10 11 12  ...
0   1
1   1  1
2   1  2  1
3   1  3  2  1
4   1  4  4  2  1
5   1  5  6  4  2  1
6   1  6  9  7  4  2  1
7   1  7 12 11  7  4  2  1
8   1  8 16 16 12  7  4  2  1
9   1  9 20 23 18 12  7  4  2  1
10  1 10 25 31 27 19 12  7  4  2  1
11  1 11 30 41 38 29 19 12  7  4  2  1
12  1 12 36 53 53 42 30 19 12  7  4  2  1
... Reformatted by _Wolfdieter Lang_, Dec 03 2012
T(5,3)=4 because the partitions of 5 in exactly 3 parts are 221 and 311, and they give rise to partitions of 4 in four ways: 221->22 and 211, 311->211 and 31, since both their Ferrers plots have 2 'mobile cells' each. [_Wouter Meeussen_, Sep 16 2010]
T(5,3) = a(6,4) = 4 because the partitions of 6 with 4 parts are 1113 and 1122, with the number of distinct parts 2 and 2, respectively, summing to 4 (see the array A103921). An example for the conjecture given as comment above. - _Wolfdieter Lang_, Dec 03 2012

V. V. Kruchinin, The number of partitions of a natural number n into parts each of which is not less than m, Math. Notes 86 (4) (2009) 505-509
R. J. Mathar, Size of the set of residues of integer powers of fixed exponent, (2017), Table 11.

Antidiagonal sums form the partition numbers (A000041).
Cf. A000070.
Cf. A008284. [Alford Arnold, Feb 07 2010]
Columns: A087811, A000601, A002621, A002622, A288341 - A288345.

Maple
```
T(n,k)=if(n
				
```

(*Needs["DiscreteMath`Combinatorica`"]; partitionexact[n_, m_] := TransposePartition /@ (Prepend[ #1, m] & ) /@ Partitions[n - m, m] *); mobile[p_?PartitionQ]:=1+Count[Drop[p,-1]-Rest[p],?Positive]; Table[Tr[mobile/@partitionexact[n,k]],{n,12},{k,n}] (* _Wouter Meeussen, Sep 16 2010 *)

T(n, k) = sum_{j=0..k} T(n-k, j), with T(n, 0) = 1 for all n>=0. A000070(n) = sum_{k=0..n} T(n, k).

O.g.f.: (1/(1-y))*(1/Product(1-x*y^k, k=1..infinity)). - Vladeta Jovovic, Jan 29 2005

Several corrections by Wolfdieter Lang, Dec 03 2012

A288344 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)...(1-x^9)).

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 138, 192, 265, 359, 482, 639, 840, 1092, 1410, 1803, 2291, 2889, 3621, 4508, 5584, 6875, 8424, 10269, 12463, 15055, 18115, 21704, 25910, 30814, 36522, 43137, 50794, 59618, 69774, 81422, 94760, 109984, 127338, 147058, 169438

Offset: 0

Seiichi Manyama, Jun 08 2017

Number of partitions of at most n into at most 9 parts.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Richard J. Mathar, Size of the Set of Residues of Integer Powers of Fixed Exponent, research paper, 2017.
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1, 0, -1, 1, -1, 1, 0, -1, 1, 2, -1, 0, 0, -1, -1, 0, 0, -1, 1, 0, 2, 0, 1, -1, 0, 0, -1, -1, 0, 0, -1, 2, 1, -1, 0, 1, -1, 1, -1, 0, -1, 0, 2, -1).

Number of partitions of at most n into at most k parts: A002621 (k=4), A002622 (k=5), A288341 (k=6), A288342 (k=7), A288343 (k=8), this sequence (k=9), A288345 (k=10).

Cf. A288256, A008638 (first differences).

x='x+O('x^99); Vec(1/((1-x)*prod(i=1, 9, (1-x^i)))) \\ Altug Alkan, Mar 28 2018

A288345 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)...(1-x^10)).

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 67, 97, 139, 194, 269, 366, 494, 658, 870, 1137, 1477, 1900, 2430, 3083, 3890, 4874, 6078, 7533, 9294, 11406, 13940, 16955, 20545, 24787, 29800, 35688, 42600, 50670, 60088, 71024, 83714, 98377, 115305, 134771, 157138, 182746, 212038

Offset: 0

Seiichi Manyama, Jun 08 2017

Number of partitions of at most n into at most 10 parts.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Richard J. Mathar, Size of the Set of Residues of Integer Powers of Fixed Exponent, research paper, 2017.
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1, 0, -1, 1, -1, 1, 0, 0, -1, 2, 0, 0, 1, -2, 0, -1, 0, 0, 0, -2, 3, 0, 1, 0, 1, 0, -1, 0, -1, 0, -3, 2, 0, 0, 0, 1, 0, 2, -1, 0, 0, -2, 1, 0, 0, -1, 1, -1, 1, 0, 1, 0, -2, 1).

Number of partitions of at most n into at most k parts: A002621 (k=4), A002622 (k=5), A288341 (k=6), A288342 (k=7), A288343 (k=8), A288344 (k=9), this sequence (k=10).

Cf. A008639 (first differences).

x='x+O('x^99); Vec(1/((1-x)*prod(i=1, 10, (1-x^i)))) \\ Altug Alkan, Mar 28 2018

A250207 The number of quartic terms in the multiplicative group modulo n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 3, 1, 5, 1, 3, 3, 1, 1, 4, 3, 9, 1, 3, 5, 11, 1, 5, 3, 9, 3, 7, 1, 15, 2, 5, 4, 3, 3, 9, 9, 3, 1, 10, 3, 21, 5, 3, 11, 23, 1, 21, 5, 4, 3, 13, 9, 5, 3, 9, 7, 29, 1, 15, 15, 9, 4, 3, 5, 33, 4, 11, 3, 35, 3, 18, 9, 5, 9, 15, 3, 39, 1

Offset: 1

R. J. Mathar, Mar 02 2015

In the character table of the multiplicative group modulo n there are phi(n) different characters. [This is made explicit for example by the number of rows in arXiv:1008.2547.] The set of the fourth powers of the characters in all representations has some cardinality, which defines the sequence.

			For n <= 6, the set of all characters in all representations consists of a subset of +1, -1, +i or -i. Their fourth powers are all +1, a single value, so a(n)=1 then.
For n=7, the set of characters is 1, -1, +-1/2 +- sqrt(3)*i/2, so their fourth powers are 1 or -1/2 +- sqrt(3)*i/2, which are three different values, so a(7)=3.
For n=11, the fourth powers of the characters may be 1, exp(+-2*i*Pi/5) or exp(+-4*i*Pi/5), which are 5 different values.

Antti Karttunen, Table of n, a(n) for n = 1..10000
R. J. Mathar, Size of the Set of Residues of Integer Powers of Fixed Exponent, (2017).
R. J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010.
Wikipedia, Dirichlet character.

Cf. A046073, A087692, A052273, A293482 - A293485.

A250207 := proc(n)
    numtheory[phi](n)/A073103(n) ;
end proc:

a[n_] := EulerPhi[n]/Count[Range[0, n-1]^4 - 1, k_ /; Divisible[k, n]];
Array[a, 80] (* Jean-François Alcover, Nov 20 2017 *)
f[p_, e_] := (p - 1)*p^(e - 1)/If[Mod[p, 4] == 1, 4, 2]; f[2, e_] := If[e <= 3, 1, 2^(e - 4)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 10 2023 *)

a(n)=my(f=factor(n)); prod(i=1,#f~, if(f[i,1]==2, 2^max(0,f[i,2]-4), f[i,1]^(f[i,2]-1)*(f[i,1]-1)/if(f[i,1]%4==1,4,2))) \\ Charles R Greathouse IV, Mar 02 2015

a(n) = A000010(n)/A073103(n).

Multiplicative with a(2^e) = 1 for e<=3; a(2^e) = 2^(e-4) for e>=4; a(p^e) = p^(e-1)*(p-1)/4 for e>=1 and p == 1 (mod 4); a(p^e) = p^(e-1)*(p-1)/2 for e>=1 and p == 3 (mod 4). (Derived from A073103.) - R. J. Mathar, Oct 13 2017

A288342 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)...(1-x^7)).

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 66, 94, 132, 181, 246, 328, 433, 564, 728, 929, 1177, 1477, 1841, 2277, 2799, 3417, 4150, 5010, 6019, 7194, 8561, 10140, 11964, 14057, 16457, 19195, 22315, 25854, 29865, 34391, 39493, 45224, 51654, 58844, 66877, 75823, 85776, 96820

Offset: 0

Seiichi Manyama, Jun 08 2017

Number of partitions of at most n into at most 7 parts.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Richard J. Mathar, Size of the Set of Residues of Integer Powers of Fixed Exponent, research paper, 2017.
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1, 0, -1, 1, -1, 0, 1, 1, 0, 1, -2, 0, 0, -2, 1, 0, 1, 1, 0, -1, 1, -1, 0, -1, 0, 2, -1).

Number of partitions of at most n into at most k parts: A002621 (k=4), A002622 (k=5), A288341 (k=6), this sequence (k=7), A288343 (k=8), A288344 (k=9), A288345 (k=10).

Cf. A288254.

x='x+O('x^99); Vec(1/((1-x)*prod(i=1, 7, (1-x^i)))) \\ Altug Alkan, Mar 28 2018

A288343 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)...(1-x^8)).

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 45, 67, 96, 136, 188, 258, 347, 463, 609, 795, 1025, 1313, 1665, 2099, 2624, 3262, 4026, 4945, 6035, 7332, 8859, 10660, 12764, 15226, 18083, 21402, 25230, 29647, 34713, 40525, 47155, 54719, 63307, 73056, 84074, 96524, 110536, 126301

Offset: 0

Seiichi Manyama, Jun 08 2017

Number of partitions of at most n into at most 8 parts.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Richard J. Mathar, Size of the Set of Residues of Integer Powers of Fixed Exponent, 2017.
Index entries for linear recurrences with constant coefficients, signature (2, 0, -1, 0, -1, 1, -1, 1, -1, 1, 1, 1, -1, -1, 1, -2, 0, -1, 1, 0, 2, -1, 1, 1, -1, -1, -1, 1, -1, 1, -1, 1, 0, 1, 0, -2, 1).

Number of partitions of at most n into at most k parts: A002621 (k=4), A002622 (k=5), A288341 (k=6), A288342 (k=7), this sequence (k=8), A288344 (k=9), A288345 (k=10).

Cf. A288255.

CoefficientList[Series[1/((1-x)Times@@(1-x^Range[8])),{x,0,50}],x] (* Harvey P. Dale, Dec 06 2017 *)

x='x+O('x^99); Vec(1/((1-x)*prod(i=1, 8, (1-x^i)))) \\ Altug Alkan, Mar 28 2018

A293483 The number of 6th powers in the multiplicative group modulo n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 5, 1, 2, 1, 2, 2, 8, 1, 3, 2, 1, 5, 11, 1, 10, 2, 3, 1, 14, 2, 5, 4, 5, 8, 2, 1, 6, 3, 2, 2, 20, 1, 7, 5, 2, 11, 23, 2, 7, 10, 8, 2, 26, 3, 10, 1, 3, 14, 29, 2, 10, 5, 1, 8, 4, 5, 11, 8, 11, 2, 35, 1, 12, 6, 10, 3, 5, 2, 13, 4, 9, 20

Offset: 1

R. J. Mathar, Oct 10 2017

The size of the set of numbers j^6 mod n, gcd(j,n)=1, 1 <= j <= n.

A000010(n) / a(n) is another multiplicative integer sequence.

R. J. Mathar, Table of n, a(n) for n = 1..10132
Richard J. Mathar, Size of the Set of Residues of Integer Powers of Fixed Exponent, 2017.

The number of k-th powers in the multiplicative group modulo n: A046073 (k=2), A087692 (k=3), A250207 (k=4), A293482 (k=5), this sequence (k=6), A293484 (k=7), A293485 (k=8).

Cf. A052275, A319100, A000010.

A293483 := proc(n)
    local r,j;
    r := {} ;
    for j from 1 to n do
        if igcd(j,n)= 1 then
            r := r union { modp(j &^ 6,n) } ;
        end if;
    end do:
    nops(r) ;
end proc:
seq(A293483(n),n=1..120) ;

a[n_] := EulerPhi[n]/Count[Range[0, n - 1]^6 - 1, k_ /; Divisible[k, n]];
Array[a, 100] (* Jean-François Alcover, May 24 2023 *)
f[p_, e_] := (p - 1)*p^(e - 1)/If[Mod[p, 6] == 1, 6, 2]; f[2, e_] := If[e <= 3, 1, 2^(e - 3)]; f[3, e_] := If[e <= 2, 1, 3^(e - 2)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 10 2023 *)

Conjecture: a(2^e) = 1 for e <= 3; a(2^e) = 2^(e-3) for e >= 3; a(3^e) = 1 for e <= 2; a(3^e) = 3^(e-2) for e >= 2; a(p^e) = (p-1)*p^(e-1)/2 for p == 5 (mod 6); a(p^e) = (p-1)*p^(e-1)/6 for p == 1 (mod 6). - R. J. Mathar, Oct 13 2017

a(n) = A000010(n)/A319100(n). This implies that the conjecture above is true. - Jianing Song, Nov 10 2019

cp's OEIS Frontend

A085311 Number of distinct 8th powers modulo n.

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A319099 Number of solutions to x^5 == 1 (mod n).

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A288341 Expansion of 1 / ((1-x)^2*(1-x^2)*(1-x^3)*...*(1-x^6)).

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A092905 Triangle, read by rows, such that the partial sums of the n-th row form the n-th diagonal, for n>=0, where each row begins with 1.

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A288344 Expansion of 1 / ((1-x)^2*(1-x^2)*(1-x^3)*...*(1-x^9)).

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A288345 Expansion of 1 / ((1-x)^2*(1-x^2)*(1-x^3)*...*(1-x^10)).

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A250207 The number of quartic terms in the multiplicative group modulo n.

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A288342 Expansion of 1 / ((1-x)^2*(1-x^2)*(1-x^3)*...*(1-x^7)).

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A288341 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)...(1-x^6)).

A288344 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)...(1-x^9)).

A288345 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)...(1-x^10)).

A288342 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)...(1-x^7)).

A288343 Expansion of 1 / ((1-x)^2(1-x^2)(1-x^3)...(1-x^8)).