A259094 -id:A259094 - cp's OEIS frontend

A053253 Coefficients of the '3rd-order' mock theta function omega(q).

1, 2, 3, 4, 6, 8, 10, 14, 18, 22, 29, 36, 44, 56, 68, 82, 101, 122, 146, 176, 210, 248, 296, 350, 410, 484, 566, 660, 772, 896, 1038, 1204, 1391, 1602, 1846, 2120, 2428, 2784, 3182, 3628, 4138, 4708, 5347, 6072, 6880, 7784, 8804, 9940, 11208, 12630

Offset: 0

Dean Hickerson, Dec 19 1999

Empirical: a(n) is the number of integer partitions mu of 2n+1 such that the diagram of mu has an odd number of cells in each row and in each column. - John M. Campbell, Apr 24 2020
From Gus Wiseman, Jun 26 2022: (Start)
By Campbell's conjecture above that a(n) is the number of partitions of 2n+1 with all odd parts and all odd conjugate parts, the a(0) = 1 through a(5) = 8 partitions are (B = 11):
  (1)  (3)    (5)      (7)        (9)          (B)
       (111)  (311)    (511)      (333)        (533)
              (11111)  (31111)    (711)        (911)
                       (1111111)  (51111)      (33311)
                                  (3111111)    (71111)
                                  (111111111)  (5111111)
                                               (311111111)
                                               (11111111111)
These partitions are ranked by A352143. (End)

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 15, 17, 31.

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Seiichi Manyama)
Steven Charlton, Explicit linear dependence congruence relations for the partition function modulo 4, arXiv:2412.17459 [math.NT], 2024. See p. 3.
Leila A. Dragonette, Some asymptotic formulas for the mock theta series of Ramanujan, Trans. Amer. Math. Soc., 72 (1952) 474-500.
John F. R. Duncan, Michael J. Griffin and Ken Ono, Proof of the Umbral Moonshine Conjecture, arXiv:1503.01472 [math.RT], 2015.
George N. Watson, The final problem: an account of the mock theta functions, J. London Math. Soc., 11 (1936) 55-80.

Other '3rd-order' mock theta functions are at A000025, A053250, A053251, A053252, A053254, A053255, A261401.
Cf. A095913(n)=a(n-3).
Cf. A259094.
Conjectured to count the partitions ranked by A352143.
A069911 = strict partitions w/ all odd parts, ranked by A258116.
A078408 = partitions w/ all odd parts, ranked by A066208.
A117958 = partitions w/ all odd parts and multiplicities, ranked by A352142.
Cf. A000009, A000290, A000701, A035363 (complement A086543), A035444, A035457, A045931, A055922, A258117.

Series[Sum[q^(2n(n+1))/Product[1-q^(2k+1), {k, 0, n}]^2, {n, 0, 6}], {q, 0, 100}]

{a(n)=local(A); if(n<0, 0, A=1+x*O(x^n); polcoeff( sum(k=0, (sqrtint(2*n+1)-1)\2, A*=(x^(4*k)/(1-x^(2*k+1))^2 +x*O(x^(n-2*(k^2-k))))), n))} /* Michael Somos, Aug 18 2006 */

{a(n)=local(A); if(n<0, 0, n++; A=1+x*O(x^n); polcoeff( sum(k=0, n-1, A*=(x/(1-x^(2*k+1)) +x*O(x^(n-k)))), n))} /* Michael Somos, Aug 18 2006 */

G.f.: omega(q) = Sum_{n>=0} q^(2*n*(n+1))/((1-q)*(1-q^3)*...*(1-q^(2*n+1)))^2.
G.f.: Sum_{k>=0} x^k/((1-x)(1-x^3)...(1-x^(2k+1))). - Michael Somos, Aug 18 2006
G.f.: (1 - G(0))/(1-x) where G(k) = 1 - 1/(1-x^(2*k+1))/(1-x/(x-1/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jan 18 2013
a(n) ~ exp(Pi*sqrt(n/3)) / (4*sqrt(n)). - Vaclav Kotesovec, Jun 10 2019
Conjectural g.f.: 1/(1 - x)*( 1 + Sum_{n >= 0} x^(3*n+1) /((1 - x)*(1 - x^3)*...*(1 - x^(2*n+1))) ). - Peter Bala, Nov 18 2024

A008672 Expansion of 1/((1-x)(1-x^3)(1-x^5)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 17, 18, 20, 22, 23, 25, 27, 29, 31, 33, 35, 37, 40, 42, 44, 47, 49, 52, 55, 57, 60, 63, 66, 69, 72, 75, 78, 82, 85, 88, 92, 95, 99, 103, 106, 110, 114, 118, 122, 126, 130, 134, 139, 143, 147, 152, 156, 161, 166

Offset: 0

N. J. A. Sloane

Number of partitions of n into odd parts less than or equal to 5.
1/((1-x^2)*(1-x^6)*(1-x^10)) is the Molien series for the icosahedral group [3,5] of order 120.
Number of partitions (d1,d2,d3) of n such that 0 <= d1/1 <= d2/2 <= d3/3. - Seiichi Manyama, Jun 04 2017

			G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 4*x^7 + 5*x^8 + 6*x^9 + 7*x^10 + ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 120, D(n;1,3,5).
W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 164 etc.
F. Hirzebruch, Letter to N. J. A. Sloane, quoted in Ges. Abh. II, 796-798.
F. Klein, Lectures on the Icosahedron ..., 2nd Rev. Ed., 1913; reprinted by Dover, NY, 1956; see pp. 236-243.
L. Smith, Polynomial Invariants of Finite Groups, Peters, 1995, p. 199 (No. 23).

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
G. E. Andrews, P. Paule, A. Riese and V. Strehl, MacMahon's partition analysis V. Bijections, recursions and magic squares
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 218
J. S. Leon, V. S. Pless and N. J. A. Sloane, Self-dual codes over GF(5), J. Combin. Theory, A 32 (1982), 178-194.
F. J. MacWilliams, C. L. Mallows and N. J. A. Sloane, Generalizations of Gleason's theorem on weight enumerators of self-dual codes, IEEE Trans. Inform. Theory, 18 (1972), 794-805; see p. 802, col. 2, foot.
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1,1,-1,0,-1,1).
Index entries for Molien series

Cf. A025799, A259094.

List([0..70], n-> Int((n^2+9*n+30)/30) ); # G. C. Greubel, Sep 08 2019

[Round((n+3)*(n+6)/30): n in [0..60]]; // Vincenzo Librandi, Jun 23 2011

seq(coeff(series(1/((1-x)*(1-x^3)*(1-x^5)), x, n+1), x, n), n = 0..70); # G. C. Greubel, Sep 08 2019

CoefficientList[Series[1/((1-x)(1-x^3)(1-x^5)),{x,0,70}],x] (* or *) LinearRecurrence[{1,0,1,-1,1,-1,0,-1,1},{1,1,1,2,2,3,4,4,5},70] (* Harvey P. Dale, Feb 07 2012 *)

{a(n) = (n^2 + 9*n)\30 + 1} /* Michael Somos, Nov 25 2002 */

[floor((n^2+9*n+30)/30) for n in (0..70)] # G. C. Greubel, Sep 08 2019

a(n) = round((n+3)*(n+6)/30).
a(n) = A025799(2n).
a(n) = floor(n^2/30 + 3*n/10 + 1). - Michael Somos, Nov 25 2002
G.f.: 1/((1-x)*(1-x^3)*(1-x^5)).
a(n) = a(-9 - n). - Michael Somos, Nov 16 2005
a(n) = a(n-1) + a(n-3) - a(n-4) + a(n-5) - a(n-6) - a(n-8) + a(n-9); a(0)=1, a(1)=1, a(2)=1, a(3)=2, a(4)=2, a(5)=3, a(6)=4, a(7)=4, a(8)=5. - Harvey P. Dale, Feb 07 2012

A008673 Expansion of 1/((1-x)(1-x^3)(1-x^5)*(1-x^7)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 7, 9, 10, 12, 14, 16, 19, 21, 24, 27, 30, 34, 38, 42, 46, 51, 56, 61, 67, 73, 79, 86, 93, 100, 108, 116, 125, 134, 143, 153, 163, 174, 185, 197, 209, 221, 235, 248, 262, 277, 292, 308, 324, 341, 358, 376, 395, 414, 434, 454, 475, 497, 519, 542, 566, 590

Offset: 0

N. J. A. Sloane

Number of partitions of n into parts 1, 3, 5, and 7. - Joerg Arndt, Jul 08 2013

Number of partitions (d1,d2,d3,d4) of n such that 0 <= d1/1 <= d2/2 <= d3/3 <= d4/4. - Seiichi Manyama, Jun 04 2017

			There are a(7)=5 partitions of n=7 into parts 1, 3, 5, and 7: (7), (511), (331), (31111), and (1111111). - _David Neil McGrath_, Feb 14 2015

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 234
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1,1,-1,1,-2,1,-1,1,-1,1,0,1,-1).

Cf. A259094.

List([0..70], n-> Int((n^3+24*n^2+171*n+630)/630) ); # G. C. Greubel, Sep 08 2019

[Floor((n^3+24*n^2+171*n+630)/630): n in [0..70]]; // G. C. Greubel, Sep 08 2019

seq(coeff(series(1/((1-x)*(1-x^3)*(1-x^5)*(1-x^7)), x, n+1), x, n), n = 0 .. 70); # G. C. Greubel, Sep 08 2019

CoefficientList[Series[1/((1-x)(1-x^3)(1-x^5)(1-x^7)), {x,0,70}], x] (* Vincenzo Librandi, Jun 22 2013 *)
LinearRecurrence[{1,0,1,-1,1,-1,1,-2,1,-1,1,-1,1,0,1,-1}, {1,1,1,2,2,3, 4,5,6,7,9,10,12,14,16,19}, 70] (* Harvey P. Dale, Jul 08 2019 *)

vector(70, n, m=n-1; (m^3+24*m^2+171*m+630)\630 ) \\ G. C. Greubel, Sep 08 2019

[floor((n^3+24*n^2+171*n+630)/630) for n in (0..70)] # G. C. Greubel, Sep 08 2019

a(n) = floor((n^3 + 24*n^2 + 171*n + 630)/630). - Tani Akinari, Jul 08 2013

a(n) = a(n-1) + a(n-3) - a(n-4) + a(n-5) - a(n-6) + a(n-7) - 2*a(n-8) + a(n-9) - a(n-10) + a(n-11) - a(n-12) + a(n-13) + a(n-15) - a(n-16). - David Neil McGrath, Feb 14 2015

A008674 Expansion of 1/((1-x)(1-x^3)(1-x^5)(1-x^7)(1-x^9)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 11, 14, 16, 19, 23, 26, 30, 35, 40, 45, 52, 58, 65, 74, 82, 91, 102, 113, 124, 138, 151, 165, 182, 198, 216, 236, 256, 277, 301, 325, 350, 379, 407, 437, 471, 504, 539, 578, 617, 658, 703, 748, 795, 847, 899, 953, 1012, 1071, 1133, 1200, 1267, 1337, 1413, 1489, 1568, 1653

Offset: 0

N. J. A. Sloane

Number of partitions of n into odd parts <= 9. - Seiichi Manyama, Jun 04 2017

Number of partitions (d1,d2,...,d5) of n such that 0 <= d1/1 <= d2/2 <= ... <= d5/5. - Seiichi Manyama, Jun 04 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 244
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1,1,-1,1,-2,2,-2,1,-2,2,-1,2,-2,2,-1,1,-1,1,-1,0,-1,1).

Cf. A259094.

R:=PowerSeriesRing(Integers(), 70); Coefficients(R!( 1/&*[1-x^(2*j+1): j in [0..4]] )); // G. C. Greubel, Sep 08 2019

seq(coeff(series(1/mul(1-x^(2*j+1), j=0..4), x, n+1), x, n), n = 0..70); # G. C. Greubel, Sep 08 2019

CoefficientList[Series[1/((1-x)(1-x^3)(1-x^5)(1-x^7)(1-x^9)), {x,0,70}], x] (* Vincenzo Librandi, Jun 22 2013 *)
LinearRecurrence[{1,0,1,-1,1,-1,1,-2,2,-2,1,-2,2,-1,2,-2,2,-1,1,-1,1,-1, 0,-1,1}, {1,1,1,2,2,3,4,5,6,8,10,11,14,16,19,23,26,30,35,40,45,52,58, 65,74}, 70] (* Harvey P. Dale, Aug 13 2016 *)

my(x='x+O('x^70)); Vec(1/prod(j=0,4,1-x^(2*j+1)) ) \\ G. C. Greubel, Sep 08 2019

a(n) = (n^4+50*n^3+855*n^2+6030*n - n*280*(n%3) +23800)\22680 \\ Hoang Xuan Thanh, Aug 12 2025

def A008674_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/prod(1-x^(2*j+1) for j in (0..4)) ).list()
A008674_list(70) # G. C. Greubel, Sep 08 2019

a(n) = floor((n^4+50*n^3+855*n^2+6030*n+23800)/22680 - n*(n mod 3)/81). - Hoang Xuan Thanh, Aug 12 2025

Typo in name fixed by Vincenzo Librandi, Jun 22 2013

A008675 Expansion of 1/( Product_{j=0..5} (1-x^(2*j+1)) ).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 17, 21, 25, 29, 34, 40, 46, 53, 62, 70, 80, 91, 103, 116, 131, 147, 164, 184, 204, 227, 252, 278, 307, 339, 372, 408, 448, 489, 534, 583, 634, 689, 749, 811, 878, 950, 1025, 1106, 1192, 1282, 1378, 1481, 1588, 1702, 1823, 1949, 2083

Offset: 0

N. J. A. Sloane

nonn

Number of partitions of n into parts 1, 3, 5, 7, 9, and 11. - Joerg Arndt, Jul 09 2013

Number of partitions (d1,d2,...,d6) of n such that 0 <= d1/1 <= d2/2 <= ... <= d6/6. - Seiichi Manyama, Jun 04 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vincenzo Librandi)
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 246
Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, -1, 1, -1, 1, -2, 2, -2, 2, -3, 2, -2, 3, -3, 3, -2, 3, -3, 3, -2, 2, -3, 2, -2, 2, -2, 1, -1, 1, -1, 1, 0, 1, -1).

Cf. A259094.

R:=PowerSeriesRing(Integers(), 65); Coefficients(R!( 1/&*[1-x^(2*j+1): j in [0..5]] )); // G. C. Greubel, Sep 08 2019

seq(coeff(series(1/mul(1-x^(2*j+1), j=0..5), x, n+1), x, n), n = 0..65); # G. C. Greubel, Sep 08 2019

CoefficientList[Series[1/((1-x)(1-x^3)(1-x^5)(1-x^7)(1-x^9)(1-x^11)), {x, 0, 65}], x] (* Vincenzo Librandi, Jun 23 2013 *)
LinearRecurrence[{1,0,1,-1,1,-1,1,-2,2,-2,2,-3,2,-2,3,-3,3,-2,3,-3,3,-2,2,-3,2,-2,2,-2,1,-1,1,-1,1,0,1,-1},{1,1,1,2,2,3,4,5,6,8,10,12,15,17,21,25,29,34,40,46,53,62,70,80,91,103,116,131,147,164,184,204,227,252,278,307},60] (* Harvey P. Dale, Oct 29 2022 *)

a(n)=(46200*((n\3+1)*[2,-1,-1][n%3+1]+[10,-4,-7][n%3+1]) +3*n^5+ 270*n^4+9005*n^3+136350*n^2+908260*n+3603600)\3742200  \\ Tani Akinari, Jul 09 2013

Vec(1/((1-x)*(1-x^3)*(1-x^5)*(1-x^7)*(1-x^9)*(1-x^11))+O(x^66)) \\ Joerg Arndt, Jul 09 2013

def A008674_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/prod(1-x^(2*j+1) for j in (0..5)) ).list()
A008674_list(65) # G. C. Greubel, Sep 08 2019

Typo in name fixed by Vincenzo Librandi, Jun 23 2013

A287997 Expansion of 1/((1-x)(1-x^3)(1-x^5) ... (1-x^13)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 26, 31, 36, 43, 50, 58, 68, 78, 90, 103, 118, 134, 153, 173, 195, 220, 247, 277, 310, 346, 385, 429, 475, 526, 582, 642, 707, 778, 854, 936, 1026, 1121, 1224, 1335, 1454, 1581, 1718, 1864, 2020, 2188, 2366, 2556

Offset: 0

Seiichi Manyama, Jun 04 2017

nonn

Number of partitions of n into odd parts less than or equal to 13.

Number of partitions (d1,d2,...,d7) of n such that 0 <= d1/1 <= d2/2 <= ... <= d7/7.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, -1, 1, -1, 1, -2, 2, -2, 2, -3, 3, -3, 3, -4, 4, -3, 4, -4, 5, -4, 4, -5, 5, -4, 4, -5, 4, -4, 3, -4, 4, -3, 3, -3, 3, -2, 2, -2, 2, -1, 1, -1, 1, -1, 0, -1, 1).

Cf. A259094.

A287998 Expansion of 1/((1-x)(1-x^3)(1-x^5) ... (1-x^15)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 37, 45, 52, 61, 72, 83, 96, 111, 128, 146, 168, 191, 217, 247, 279, 314, 355, 398, 446, 501, 558, 622, 693, 770, 853, 946, 1045, 1153, 1273, 1400, 1538, 1690, 1852, 2027, 2219, 2422, 2642, 2881, 3136, 3409

Offset: 0

Seiichi Manyama, Jun 04 2017

nonn

Number of partitions of n into odd parts less than or equal to 15.

Number of partitions (d1,d2,...,d8) of n such that 0 <= d1/1 <= d2/2 <= ... <= d8/8.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, -1, 1, -1, 1, -2, 2, -2, 2, -3, 3, -3, 4, -5, 4, -4, 5, -5, 6, -5, 6, -7, 7, -6, 7, -8, 7, -7, 7, -8, 7, -7, 7, -8, 7, -6, 7, -7, 6, -5, 6, -5, 5, -4, 4, -5, 4, -3, 3, -3, 2, -2, 2, -2, 1, -1, 1, -1, 1, 0, 1, -1).

Cf. A259094.

A288000 Expansion of 1/((1-x)(1-x^3)(1-x^5) ... (1-x^17)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 53, 63, 74, 86, 100, 116, 134, 154, 178, 203, 232, 265, 301, 341, 387, 436, 492, 554, 621, 696, 779, 870, 969, 1080, 1199, 1331, 1476, 1632, 1803, 1991, 2193, 2414, 2655, 2914, 3196, 3502, 3832

Offset: 0

Seiichi Manyama, Jun 04 2017

nonn

Number of partitions of n into odd parts less than or equal to 17.

Number of partitions (d1,d2,...,d9) of n such that 0 <= d1/1 <= d2/2 <= ... <= d9/9.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, -1, 1, -1, 1, -2, 2, -2, 2, -3, 3, -3, 4, -5, 5, -5, 5, -6, 7, -6, 7, -8, 9, -8, 9, -10, 10, -10, 10, -12, 12, -11, 11, -13, 12, -12, 12, -13, 13, -12, 12, -12, 13, -11, 11, -12, 12, -10, 10, -10, 10, -9, 8, -9, 8, -7, 6, -7, 6, -5, 5, -5, 5, -4, 3, -3, 3, -2, 2, -2, 2, -1, 1, -1, 1, -1, 0, -1, 1).

Cf. A259094.

A288001 Expansion of 1/((1-x)(1-x^3)(1-x^5) ... (1-x^19)).

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64, 75, 88, 102, 119, 138, 159, 184, 211, 242, 277, 316, 359, 409, 463, 524, 592, 667, 750, 843, 945, 1057, 1182, 1318, 1469, 1635, 1816, 2014, 2233, 2470, 2730, 3014, 3323, 3659, 4026

Offset: 0

Seiichi Manyama, Jun 04 2017

nonn

Number of partitions of n into odd parts less than or equal to 19.

Number of partitions (d1,d2,...,d10) of n such that 0 <= d1/1 <= d2/2 <= ... <= d10/10.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1, 0, 1, -1, 1, -1, 1, -2, 2, -2, 2, -3, 3, -3, 4, -5, 5, -5, 6, -7, 7, -7, 8, -9, 10, -9, 11, -12, 12, -12, 13, -15, 15, -15, 16, -18, 17, -17, 18, -20, 19, -19, 20, -21, 21, -20, 21, -22, 22, -20, 22, -22, 21, -20, 21, -21, 20, -19, 19, -20, 18, -17, 17, -18, 16, -15, 15, -15, 13, -12, 12, -12, 11, -9, 10, -9, 8, -7, 7, -7, 6, -5, 5, -5, 4, -3, 3, -3, 2, -2, 2, -2, 1, -1, 1, -1, 1, 0, 1, -1).

Cf. A259094.

CoefficientList[Series[1/Times@@Table[(1-x^exp),{exp,Range[1,19,2]}],{x,0,60}],x] (* Harvey P. Dale, Mar 08 2025 *)

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A053253 Coefficients of the '3rd-order' mock theta function omega(q).

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A008672 Expansion of 1/((1-x)*(1-x^3)*(1-x^5)).

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A008673 Expansion of 1/((1-x)*(1-x^3)*(1-x^5)*(1-x^7)).

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A008674 Expansion of 1/((1-x)*(1-x^3)*(1-x^5)*(1-x^7)*(1-x^9)).

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A008675 Expansion of 1/( Product_{j=0..5} (1-x^(2*j+1)) ).

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A287997 Expansion of 1/((1-x)(1-x^3)(1-x^5) ... (1-x^13)).

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A008672 Expansion of 1/((1-x)(1-x^3)(1-x^5)).

A008673 Expansion of 1/((1-x)(1-x^3)(1-x^5)*(1-x^7)).

A008674 Expansion of 1/((1-x)(1-x^3)(1-x^5)(1-x^7)(1-x^9)).