A006950 -id:A006950 - cp's OEIS frontend

A295832 Expansion of Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k)))^k.

1, 1, 1, 3, 5, 8, 12, 20, 33, 50, 74, 114, 175, 257, 375, 555, 814, 1171, 1677, 2406, 3435, 4855, 6825, 9591, 13428, 18667, 25851, 35745, 49250, 67544, 92340, 125966, 171345, 232257, 313945, 423470, 569778, 764465, 1023231, 1366827, 1821756, 2422394, 3214318, 4257088, 5627086, 7422941

Offset: 0

Ilya Gutkovskiy, Nov 28 2017

nonn

Cf. A000219, A006950, A113415, A156616, A224364, A263140, A273225, A273226, A274621, A295831.

nmax = 45; CoefficientList[Series[Product[((1 + x^(2 k - 1))/(1 - x^(2 k)))^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[Exp[Sum[x^k ((-1)^(k + 1) + x^k)/(k (1 - x^(2 k))^2), {k, 1, nmax}]], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + x^(2*k-1))/(1 - x^(2*k)))^k.
G.f.: exp(Sum_{k>=1} x^k*((-1)^(k+1) + x^k)/(k*(1 - x^(2*k))^2)).
a(n) ~ exp(3 * (7*Zeta(3))^(1/3) * n^(2/3) / 4 + Pi^2 * n^(1/3) / (24 * (7*Zeta(3))^(1/3)) - Pi^4 / (12096 * Zeta(3)) + 1/12) * (7*Zeta(3))^(7/36) / (A * 2^(23/24) * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Nov 28 2017

A340623 The number of partitions of n without repeated odd parts having more even parts than odd parts.

Original entry on oeis.org

0, 0, 1, 0, 2, 1, 3, 3, 5, 7, 8, 13, 14, 23, 23, 37, 39, 59, 63, 90, 101, 136, 156, 201, 239, 296, 355, 428, 523, 617, 754, 878, 1078, 1243, 1517, 1741, 2121, 2426, 2928, 3348, 4021, 4596, 5468, 6257, 7400, 8472, 9936, 11389, 13285, 15233, 17645, 20244, 23346

Offset: 0

Jeremy Lovejoy, Jan 13 2021

nonn

			a(7) = 3 counts the partitions [4,2,1], [3,2,2], and [2,2,2,1].

Seiichi Manyama, Table of n, a(n) for n = 0..10000
B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

Cf. A006950, A340621, A340622, A340647.

b:= proc(n, i, c) option remember; `if`(n=0,
      `if`(c<0, 1, 0), `if`(i<1, 0, (t-> add(b(n-i*j, i-1, c+j*
      `if`(t, 1, -1)), j=0..min(n/i, `if`(t, 1, n))))(i::odd)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 13 2021

nmax = 60; CoefficientList[Series[Product[(1 + x^(2*k - 1))/(1 - x^(2*k)), {k, 1, nmax/2}] - Sum[x^(k^2)/Product[(1 - x^(2*j))^2, {j, 1, k}], {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 14 2021 *)

my(N=66, x='x+O('x^N)); concat([0, 0], Vec(prod(k=1, N, (1+x^(2*k-1))/(1-x^(2*k)))-sum(k=0, sqrt(N), x^(k^2)/prod(j=1, k, (1-x^(2*j))^2)))) \\ Seiichi Manyama, Jan 14 2021

G.f.: (Product_{k>=1} (1+q^(2*k-1))/(1-q^(2*k))) - Sum_{n>=0} q^(n^2)/Product_{k=1..n} (1-q^(2*k))^2.

A085642 Number of columns in the character table of the symmetric group S_n that have zero sum.

Original entry on oeis.org

0, 1, 1, 2, 3, 6, 8, 12, 17, 26, 35, 49, 66, 92, 121, 161, 211, 280, 360, 466, 596, 766, 968, 1225, 1538, 1935, 2408, 2996, 3707, 4588, 5636, 6918, 8456, 10329, 12552, 15236, 18431, 22275, 26817, 32242, 38661, 46306, 55294, 65942, 78464, 93252, 110561

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 11 2003

nonn

Conjecture: Equals the number of partitions of n with at least one part congruent to 2 mod 4. - Vladeta Jovovic, Jul 12 2003. This conjecture was established by Christine Bessenrodt and Jorn B. Olsson (olsson(AT)math.ku.dk), Sep 13 2004.

Also number of partitions of n with some odd part repeated. - Vladeta Jovovic, Feb 05 2005

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000
Arvind Ayyer, Hiranya Kishore Dey, and Digjoy Paul, How large is the character degree sum compared to the character table sum for a finite group?, arXiv preprint arXiv:2406.06036 [math.RT], 2024. See p. 6.
Arvind Ayyer, Hiranya Kishore Dey, and Digjoy Paul, On the sum of the entries in a character table, Proc. 36th Conf. Formal Power Series Alg. Comb., Sem. Lotharingien Comb (2024) Vol. 91B, Art. No. 99.
Christine Bessenrodt and Jorn Olsson, On the sequence A085642
Dominique Gouyou-Beauchamps and Philippe Nadeau, Signed Enumeration of Ribbon Tableaux with Local Rules and Generalizations of the Schensted Correspondence, in Formal Power Series and Algebraic Combinatorics, Nankai University, Tianjin, China, 2007.
Dominique Gouyou-Beauchamps and Philippe Nadeau, Signed enumeration of ribbon tableaux: an approach through growth diagrams, Journal of Algebraic Combinatorics, 2011; DOI 10.1007/s10801-011-0324-2.

Cf. A006950, A082733.

Rest[PartitionsP[Range[0,47]] - CoefficientList[Series[Product[(1+x^(2 k - 1))/(1 - x^(2 k)), {k,48}], {x,0,47}], x]] (* Wouter Meeussen, Dec 20 2017 *)

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)). - Vaclav Kotesovec, Jul 11 2018

Corrected and extended by Vladeta Jovovic, Jul 12 2003

More terms from David Wasserman, Feb 08 2005

A131945 Number of partitions of n where odd parts are distinct or repeated once.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 8, 10, 15, 18, 26, 32, 45, 55, 74, 90, 119, 145, 188, 228, 291, 351, 442, 532, 664, 796, 982, 1172, 1435, 1708, 2076, 2462, 2972, 3512, 4214, 4966, 5929, 6965, 8272, 9688, 11457, 13383, 15762, 18362, 21543, 25031, 29264, 33922, 39533, 45717

Offset: 0

Brian Drake, Jul 30 2007

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Also number of partitions of n such that every part is not congruent to 3 mod 6. More generally, g.f. for number of partitions of n such that every odd part occurs at most m times is product_{n=1..oo} (1-q^((m+1)*(2*n-1)))/(1-q^n). Similarly, g.f. for number of partitions of n such that every even part occurs at most m times is product_{n=1..oo} (1-q^((2*m+2)*n))/(1-q^n). - Vladeta Jovovic, Aug 01 2007

			a(6) = 8 because we have 6, 5+1, 4+2, 4+1+1, 3+3, 3+2+1, 2+2+2 and 2+2+1+1. The three excluded partitions of 6 are 3+1+1+1, 2+1+1+1+1 and 1+1+1+1+1+1.
G.f. = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 5*x^5 + 8*x^6 + 10*x^7 + 15*x^8 + ...
G.f. = 1/q + q^5 + 2*q^11 + 2*q^17 + 4*q^23 + 5*q^29 + 8*q^35 + 10*q^41 + ...

Brian Drake and Seiichi Manyama, Table of n, a(n) for n = 0..1000 (first 101 terms from Brian Drake)
Brian Drake, Limits of areas under lattice paths, Discrete Math. 309 (2009), no. 12, 3936-3953.
Mircea Merca, Overpartitions in terms of 2-adic valuation, Aequat. Math. (2024). See p. 11.
James A. Sellers, Elementary Proofs of Two Congruences for Partitions with Odd Parts Repeated at Most Twice, arXiv:2409.12321 [math.NT], 2024. See p. 2.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000122, A000700, A006950, A010054, A121373, A131942, A279320.

A:= series(product( (1-q^(6*n-3))/(1-q^n), n=1..20),q,21): seq(coeff(A,q,i), i=0..20);

a[ n_] := SeriesCoefficient[ QPochhammer[ x^3] / (QPochhammer[ x] QPochhammer[ x^6]), {x, 0, n}]; (* Michael Somos, Jan 29 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3, x^6] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 11 2015 *)
nmax = 50; CoefficientList[Series[Product[1 / ((1-x^k) * (1+x^(3*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 11 2016 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)), n))}; /* Michael Somos, Aug 05 2007 */

G.f.: product_{n=1..oo} (1-q^(6n-3))/(1-q^n).
Expansion of chi(-x^3) / f(-x) in powers of x where chi(), f() are Ramanujan theta functions. - Michael Somos, Aug 05 2007
Expansion of q^(1/6) * eta(q^3) / (eta(q) * eta(q^6)) in powers of q. - Michael Somos, Aug 05 2007
Euler transform of period 6 sequence [ 1, 1, 0, 1, 1, 1, ...]. - Michael Somos, Aug 05 2007
a(n) ~ sqrt(5) * exp(sqrt(5*n)*Pi/3) / (12*n). - Vaclav Kotesovec, Dec 11 2016

A261384 Expansion of Product_{k>=1} (1+x^k)^(2k-1) / (1-x^k)^(2k).

Original entry on oeis.org

1, 3, 12, 39, 117, 331, 893, 2307, 5766, 13986, 33046, 76302, 172567, 383013, 835731, 1795236, 3801105, 7941439, 16386777, 33423342, 67435311, 134675784, 266385932, 522135379, 1014643823, 1955656848, 3740191268, 7100290646, 13383997996, 25058666367

Offset: 0

Vaclav Kotesovec, Aug 17 2015

nonn

Convolution of A161870 and A255835.

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000

Cf. A006950, A015128, A156616, A161870, A255835, A261386.

nmax = 40; CoefficientList[Series[Product[(1+x^k)^(2*k-1)/(1-x^k)^(2*k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ (7*Zeta(3))^(2/9) * exp(1/6 - Pi^4/(6048*Zeta(3)) - Pi^2 * n^(1/3) / (12*(7*Zeta(3))^(1/3)) + 3/2*(7*Zeta(3))^(1/3) * n^(2/3)) / (A^2 * 2^(1/6) * sqrt(3*Pi) * n^(13/18)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

A308806 Expansion of 1 / Sum_{k>=0} (-x)^(k(3k - 1)/2).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 8, 11, 14, 18, 23, 30, 40, 52, 67, 86, 111, 145, 188, 243, 314, 406, 527, 683, 883, 1141, 1475, 1910, 2474, 3201, 4140, 5355, 6929, 8968, 11603, 15009, 19416, 25121, 32507, 42060, 54413, 70393, 91071, 117831, 152453, 197238, 255175, 330137, 427130, 552620

Offset: 0

Ilya Gutkovskiy, Jun 25 2019

nonn

Cf. A000041, A000326, A006950, A181324, A208061, A317665.

nmax = 53; CoefficientList[Series[1/Sum[(-x)^(k (3 k - 1)/2), {k, 0, nmax}], {x, 0, nmax}], x]

G.f.: 1 / Sum_{k>=0} (-x)^A000326(k).

A340621 The number of partitions of n without repeated odd parts having more odd parts than even parts.

Original entry on oeis.org

0, 1, 0, 1, 1, 1, 2, 1, 4, 2, 6, 3, 9, 6, 12, 10, 17, 17, 22, 26, 30, 40, 40, 57, 55, 82, 74, 112, 103, 153, 140, 203, 193, 270, 262, 351, 357, 458, 478, 589, 641, 760, 846, 971, 1114, 1244, 1450, 1582, 1880, 2018, 2412, 2558, 3086, 3247, 3914, 4102, 4949

Offset: 0

Jeremy Lovejoy, Jan 13 2021

nonn

			a(8) = 4 counts the partitions [7,1], [5,3], [5,2,1], and [4,3,1].

Seiichi Manyama, Table of n, a(n) for n = 0..10000
B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

Cf. A006950, A143184, A340622, A340623.

b:= proc(n, i, c) option remember; `if`(n=0,
      `if`(c>0, 1, 0), `if`(i<1, 0, (t-> add(b(n-i*j, i-1, c+j*
      `if`(t, 1, -1)), j=0..min(n/i, `if`(t, 1, n))))(i::odd)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 13 2021

b[n_, i_, c_] := b[n, i, c] = If[n == 0,
     If[c > 0, 1, 0], If[i < 1, 0, Function[t, Sum[b[n - i*j, i - 1, c + j*
     If[t, 1, -1]], {j, 0, Min[n/i, If[t, 1, n]]}]][OddQ[i]]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 29 2022, after Alois P. Heinz *)

my(N=66, x='x+O('x^N)); concat(0, Vec(sum(k=1, sqrt(N), x^(k^2)*(1-x^(2*k))/prod(j=1, k, (1-x^(2*j))^2)))) \\ Seiichi Manyama, Jan 14 2021

G.f.: Sum_{n>=1} q^(n^2)*(1-q^(2*n))/Product_{k=1..n} (1-q^(2*k))^2.

a(n) ~ exp(Pi*sqrt(2*n/5)) / (2^(3/2) * 5^(3/4) * n). - Vaclav Kotesovec, Jan 14 2021

A340622 The number of partitions of n without repeated odd parts having an equal number of odd parts and even parts.

Original entry on oeis.org

1, 0, 0, 1, 0, 2, 0, 3, 1, 4, 2, 5, 5, 6, 8, 8, 14, 10, 20, 14, 30, 20, 40, 29, 56, 42, 72, 62, 96, 88, 122, 125, 160, 174, 202, 239, 263, 322, 334, 431, 434, 566, 554, 739, 719, 954, 920, 1222, 1192, 1552, 1524, 1964, 1962, 2466, 2500, 3088, 3196, 3848, 4046

Offset: 0

Jeremy Lovejoy, Jan 13 2021

nonn

			a(9) = 4 counts the partitions [8,1], [7,2], [6,3], and [5,4].

Vaclav Kotesovec, Table of n, a(n) for n = 0..6000
B. Kim, E. Kim, and J. Lovejoy, Parity bias in partitions, European J. Combin., 89 (2020), 103159, 19 pp.

Cf. A006950, A340621, A340623.

b:= proc(n, i, c) option remember; `if`(n=0,
      `if`(c=0, 1, 0), `if`(i<1, 0, (t-> add(b(n-i*j, i-1, c+j*
      `if`(t, 1, -1)), j=0..min(n/i, `if`(t, 1, n))))(i::odd)))
    end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..60);  # Alois P. Heinz, Jan 13 2021

b[n_, i_, c_] := b[n, i, c] = If[n == 0,
     If[c == 0, 1, 0], If[i < 1, 0, Function[t, Sum[b[n-i*j, i-1, c + j*
     If[t, 1, -1]], {j, 0, Min[n/i, If[t, 1, n]]}]][OddQ[i]]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, May 29 2022, after Alois P. Heinz *)

G.f.: Sum_{n>=1} q^(n^2+2*n)/Product_{k=1..n} (1-q^(2*k))^2.

a(n) ~ exp(Pi*sqrt(2*n/5)) / (2^(1/2) * 5^(3/4) * (1 + sqrt(5)) * n). - Vaclav Kotesovec, Jan 14 2021

A346797 Number of partitions of n into parts congruent to 0, 2 or 5 (mod 7).

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 1, 2, 1, 3, 2, 3, 4, 3, 7, 4, 9, 6, 10, 11, 11, 17, 13, 22, 19, 25, 29, 28, 42, 34, 53, 46, 61, 67, 69, 92, 83, 115, 109, 133, 149, 152, 198, 182, 243, 233, 282, 309, 324, 398, 385, 485, 483, 563, 621, 648, 784, 768, 944, 947, 1096, 1194, 1262

Offset: 0

Ludovic Schwob, Aug 04 2021

nonn

			For n=17 the a(17)=6 solutions are 2+2+2+2+2+2+5, 2+2+2+2+2+7, 2+2+2+2+9, 2+5+5+5, 5+5+7 and 5+12.

Ludovic Schwob, Table of n, a(n) for n = 0..10000

Cf. A000041, A047386, A274830, A006950, A036820, A036822, A195848, A035363, A195849, A346798.

CoefficientList[Series[Product[1/((1 - x^(7*k))(1 - x^(7*k-2))(1 - x^(7*k-5))),{k,52}],{x,0,52}],x] (* Stefano Spezia, Aug 04 2021 *)

G.f.: Product_{k>=1} 1/((1 - x^(7*k))*(1 - x^(7*k-2))*(1 - x^(7*k-5))).
a(n) = a(n-2) + a(n-5) - a(n-11) - a(n-17) + + - - (with a(0)=1 and a(n) = 0 for negative n), where 2, 5, 11, 17, ... is the sequence A274830.
a(n) ~ exp(Pi*sqrt(2*n/7)) / (8*cos(3*Pi/14)*n). - Vaclav Kotesovec, Aug 05 2021

A346798 Number of partitions of n into parts congruent to 0, 3 or 4 (mod 7).

Original entry on oeis.org

1, 0, 0, 1, 1, 0, 1, 2, 1, 1, 3, 3, 2, 3, 6, 4, 4, 8, 9, 6, 10, 15, 12, 12, 21, 22, 18, 25, 36, 30, 32, 48, 52, 45, 60, 78, 72, 75, 105, 113, 105, 130, 166, 156, 166, 218, 236, 224, 274, 332, 325, 345, 436, 469, 462, 544, 649, 644, 688, 839, 907, 903, 1051

Offset: 0

Ludovic Schwob, Aug 04 2021

nonn

			For n=19 the a(19)=6 solutions are 3+3+3+3+3+4, 3+3+3+3+7, 3+3+3+10, 3+4+4+4+4, 4+4+4+7, and 4+4+11.

Ludovic Schwob, Table of n, a(n) for n = 0..10000

Cf. A000041, A047342, A057570, A006950, A036820, A036822, A195848, A035363, A195849, A346797.

CoefficientList[Series[Product[1/((1 - x^(7*k))(1 - x^(7*k-3))(1 - x^(7*k-4))),{k,55}],{x,0,55}],x] (* Stefano Spezia, Aug 04 2021 *)

G.f.: Product_{k>=1} 1/((1 - x^(7*k))*(1 - x^(7*k-3))*(1 - x^(7*k-4))).
a(n) = a(n-3) + a(n-4) - a(n-13) - a(n-15) + + - - (with a(0)=1 and a(n) = 0 for negative n), where 3, 4, 13, 15, ... is the sequence A057570.
a(n) ~ exp(Pi*sqrt(2*n/7)) / (8*cos(Pi/14)*n). - Vaclav Kotesovec, Aug 05 2021

cp's OEIS Frontend

A295832 Expansion of Product_{k>=1} ((1 + x^(2*k-1))/(1 - x^(2*k)))^k.

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A340623 The number of partitions of n without repeated odd parts having more even parts than odd parts.

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A085642 Number of columns in the character table of the symmetric group S_n that have zero sum.

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A131945 Number of partitions of n where odd parts are distinct or repeated once.

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A261384 Expansion of Product_{k>=1} (1+x^k)^(2*k-1) / (1-x^k)^(2*k).

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A308806 Expansion of 1 / Sum_{k>=0} (-x)^(k*(3*k - 1)/2).

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A340621 The number of partitions of n without repeated odd parts having more odd parts than even parts.

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A340622 The number of partitions of n without repeated odd parts having an equal number of odd parts and even parts.

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A295832 Expansion of Product_{k>=1} ((1 + x^(2k-1))/(1 - x^(2k)))^k.

A261384 Expansion of Product_{k>=1} (1+x^k)^(2k-1) / (1-x^k)^(2k).

A308806 Expansion of 1 / Sum_{k>=0} (-x)^(k(3k - 1)/2).