A035959 -id:A035959 - cp's OEIS frontend

A328546 Number of 12-regular partitions of n (no part is a multiple of 12).

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 76, 100, 133, 173, 226, 290, 374, 475, 605, 762, 960, 1199, 1497, 1856, 2299, 2831, 3482, 4261, 5208, 6337, 7700, 9321, 11266, 13572, 16325, 19578, 23444, 27999, 33389, 39721, 47185, 55929, 66199, 78199, 92246

Offset: 0

N. J. A. Sloane, Oct 19 2019

nonn

Kathiravan, T., and S. N. Fathima. "On L-regular bipartitions modulo L." The Ramanujan Journal 44.3 (2017): 549-558.

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

f:=(k,M) -> mul(1-q^(k*j),j=1..M);
LRP := (L,M) -> f(L,M)/f(1,M);
s := L -> seriestolist(series(LRP(L,80),q,60));
s(12);

Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 12], 0, 2] ], {n, 0, 46}] (* Robert Price, Jul 28 2020 *)

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=12. - Vaclav Kotesovec, Aug 01 2022

A286653 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, 2, 2, 0, 1, 1, 2, 2, 2, 0, 1, 1, 2, 3, 4, 3, 0, 1, 1, 2, 3, 4, 5, 4, 0, 1, 1, 2, 3, 5, 6, 7, 5, 0, 1, 1, 2, 3, 5, 6, 9, 9, 6, 0, 1, 1, 2, 3, 5, 7, 10, 12, 13, 8, 0, 1, 1, 2, 3, 5, 7, 10, 13, 16, 16, 10, 0, 1, 1, 2, 3, 5, 7, 11, 14, 19, 22, 22, 12, 0

Offset: 0

Ilya Gutkovskiy, May 11 2017

A(n,k) is the number of partitions of n in which no parts are multiples of k.

A(n,k) is also the number of partitions of n into at most k-1 copies of each part.

			Square array begins:
  1,  1,  1,  1,  1,  1,  ...
  0,  1,  1,  1,  1,  1,  ...
  0,  1,  2,  2,  2,  2,  ...
  0,  2,  2,  3,  3,  3,  ...
  0,  2,  4,  4,  5,  5,  ...
  0,  3,  5,  6,  6,  7,  ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened
Eric Weisstein's World of Mathematics, Partition Function b_k
Index entries for sequences related to partitions

Columns k=1-13 give: A000007, A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546, A341714.
Main diagonal gives A000041.
Mirror of A061198.
Cf. A061199, A210485.

b:= proc(n, i, k) option remember; `if`(n=0, [1, 0], `if`(k*i*(i+1)/2[0, l[1]*j]+l)(b(n-i*j, i-1, k)), j=0..min(n/i, k))))
    end:
A:= (n, k)-> b(n$2, k-1)[1]:
seq(seq(A(n, 1+d-n), n=0..d), d=0..16);  # Alois P. Heinz, Oct 17 2018

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

A121591 Expansion of (eta(q^5) / eta(q))^6 in powers of q.

Original entry on oeis.org

1, 6, 27, 98, 315, 912, 2456, 6210, 14937, 34390, 76317, 163896, 342062, 695736, 1382880, 2691586, 5139906, 9644622, 17808040, 32393370, 58113312, 102914152, 180062622, 311488920, 533124225, 903324372, 1516110165, 2521780688, 4158863310, 6803237280, 11043320922, 17794350786

Offset: 1

Michael Somos, Aug 09 2006

nonn

			G.f. = q + 6*q^2 + 27*q^3 + 98*q^4 + 315*q^5 + 912*q^6 + 2456*q^7 + 6210*q^8 + ...

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.

Cf. A035959, A106248.

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^5] / QPochhammer[ q])^6, {q, 0, n}]; (* Michael Somos, May 22 2013 *)
nmax = 40; Rest[CoefficientList[Series[x * Product[((1 - x^(5*k)) / (1 - x^k))^6, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 07 2015 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^5 + A) / eta(x + A))^6, n))};

Euler transform of period 5 sequence [6, 6, 6, 6, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u*v * (1 + 125 * u*v) - (u+v) * (u^2 - 13 * u*v + v^2). - Michael Somos, May 22 2013
G.f. is a period 1 Fourier series which satisfies f(-1 / (5 t)) = 1/125 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A106248. - Michael Somos, May 22 2013
G.f.: x * (Product_{k>0} (1 - x^(5*k)) / (1 - x^k))^6.
Convolution inverse of A106248, 6th power of A035959. - Michael Somos, Aug 09 2015
a(n) ~ exp(4*Pi*sqrt(n/5)) / (125 * sqrt(2) * 5^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
a(1) = 1, a(n) = (6/(n-1))*Sum_{k=1..n-1} A116073(k)*a(n-k) for n > 1. - Seiichi Manyama, Mar 31 2017

A145466 Expansion of q^(1/6) * eta(q) / eta(q^5) in powers of q.

Original entry on oeis.org

1, -1, -1, 0, 0, 2, -1, 0, 0, 0, 3, -2, -2, 0, 0, 4, -3, -2, 0, 0, 7, -5, -3, 0, 0, 10, -6, -4, 0, 0, 15, -10, -7, 0, 0, 20, -13, -8, 0, 0, 28, -19, -13, 0, 0, 38, -25, -16, 0, 0, 52, -34, -23, 0, 0, 68, -44, -28, 0, 0, 91, -60, -40, 0, 0, 118, -76, -48, 0, 0, 153, -100, -66, 0, 0, 196, -127, -82, 0, 0, 252, -164, -107, 0, 0

Offset: 0

Michael Somos, Oct 11 2008

sign

			G.f. = 1 - x - x^2 + 2*x^5 - x^6 + 3*x^10 - 2*x^11 - 2*x^12 + 4*x^15 + ...
G.f. = 1/q - q^5 - q^11 + 2*q^29 - q^35 + 3*q^59 - 2*q^65 - 2*q^71 + 4*q^89 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A035969, A058511, A145467, A145468.

a[ n_] := SeriesCoefficient[ QPochhammer[ x] / QPochhammer[ x^5], {x, 0, n}]; (* Michael Somos, Jun 26 2014 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^5 + A), n))};

Expansion of 1 / (G(x) * H(x)) = G(x^5)^2 - x * G(x^5) * H(x^5) - x^2 * H(x^5)^2 in powers of x where G(), H() are the Rogers-Ramanujan functions.
Euler transform of period 5 sequence [ -1, -1, -1, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (180 t)) = 5^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A035959.
Given g.f. A(x), then B(x) = A(x^3)^2 / x satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^3 + v^3 - 5*u*v - u^2*v^2.
Given g.f. A(x), then B(x) = A(x^3)^2 / x satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v * u^2 * w^2 + 5 * u * w * (u + w) - v^2 * (u^2 + u*w + w^2).
a(5*n + 3) = a(5*n + 4) = 0.
G.f.: 1 / (Product_{k>0} P(5, x^k)) where P(n,x) is the n-th cyclotomic polynomial.
a(5*n) = A145467(n). a(5*n + 1) = - A035969(n). a(5*n + 2) = - A145468(n).
Convolution inverse of A035959.
a(n) = -(1/n)*Sum_{k=1..n} A116073(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

A061199 Upper right triangle read by columns where T(n,k), with k >= n, is the 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, 2, 0, 2, 2, 3, 0, 2, 4, 4, 5, 0, 3, 5, 6, 6, 7, 0, 4, 7, 9, 10, 10, 11, 0, 5, 9, 12, 13, 14, 14, 15, 0, 6, 13, 16, 19, 20, 21, 21, 22, 0, 8, 16, 22, 25, 27, 28, 29, 29, 30, 0, 10, 22, 29, 34, 37, 39, 40, 41, 41, 42, 0, 12, 27, 38, 44, 49, 51, 53, 54, 55, 55, 56, 0, 15, 36

Offset: 0

Henry Bottomley, Apr 20 2001

			T(2,4) = 4 since the possible partitions of 4 are on the first definition (no term more than twice) 1+1+2, 2+2, 1+3, or 4 and on the second definition (no term a multiple of 3) 1+1+1+1, 1+1+2, 2+2, or 4.
Triangle T(n,k) begins:
1, 0, 0, 0, 0, 0,  0,  0,  0,  0, ...
   1, 1, 2, 2, 3,  4,  5,  6,  8, ...
      2, 2, 4, 5,  7,  9, 13, 16, ...
         3, 4, 6,  9, 12, 16, 22, ...
            5, 6, 10, 13, 19, 25, ...
               7, 10, 14, 20, 27, ...
                  11, 14, 21, 28, ...
                      15, 21, 29, ...
                          22, 29, ...
                              30, ...

Alois P. Heinz, Columns k = 0..140, flattened

Rows effectively include A000007, A000009, A000726, A001935, A035959.
Main diagonal is A000041.
A061198 is the same table but includes cases where n>k.
T(n,2*n) gives: A232623.

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:
T:= (n, k)-> b(k$2, n):
seq(seq(T(n, k), n=0..k), k=0..12);  # Alois P. Heinz, Nov 27 2013

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]}]]]; T[n_, k_] := b[k, k, n]; Table[Table[T[n, k], {n, 0, k}], {k, 0, 12}] // Flatten (* Jean-François Alcover, Jan 28 2015, after Alois P. Heinz *)

A107234 Expansion of 1 / Product_{n>=0} (1-q^(5n+1))(1-q^(5n+2))(1-q^(5n+3)).

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 8, 10, 14, 18, 23, 29, 38, 47, 60, 74, 92, 112, 139, 168, 205, 247, 298, 356, 429, 509, 607, 718, 850, 1000, 1180, 1381, 1620, 1890, 2206, 2564, 2983, 3453, 4000, 4618, 5330, 6133, 7059, 8097, 9289, 10630, 12159, 13877

Offset: 0

Ralf Stephan, May 13 2005

nonn

a(n) is the number of partitions of n into parts 5k+1, 5k+2 or 5k+3. - George Beck, Aug 09 2020

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

Cf. A035959, A107235, A107236, A107237.

nmax = 50; CoefficientList[Series[1/Product[(1 - x^(5*k+1))*(1 - x^(5*k+2))*(1 - x^(5*k+3)), {k, 0, nmax/5}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 07 2021 *)

a(n) ~ Pi^(1/5) * exp(Pi*sqrt(2*n/5)) / (Gamma(4/5) * 2^(3/5) * 5^(9/10) * n^(3/5)). - Vaclav Kotesovec, Jan 07 2021

A107235 Expansion of 1 / Product_{n>=0} (1 - q^(5n+1))(1 - q^(5n+2))(1 - q^(5n+4)).

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 7, 8, 12, 14, 19, 23, 31, 37, 48, 57, 73, 86, 109, 128, 159, 187, 229, 269, 326, 382, 458, 535, 638, 742, 879, 1019, 1200, 1388, 1625, 1875, 2185, 2514, 2916, 3347, 3868, 4427, 5099, 5822, 6683, 7614, 8712, 9904, 11301, 12821, 14589

Offset: 0

Ralf Stephan, May 13 2005

nonn

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

Cf. A035959, A107234, A107236, A107237.

nmax = 50; CoefficientList[Series[1/Product[(1 - x^(5*k+1))*(1 - x^(5*k+2))*(1 - x^(5*k+4)), {k, 0, nmax/5}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 07 2021 *)

a(n) ~ Pi^(2/5) * exp(Pi*sqrt(2*n/5)) / (Gamma(3/5) * 2^(7/10) * 5^(4/5) * n^(7/10)). - Vaclav Kotesovec, Jan 07 2021

A107236 Expansion of 1 / Product_{n>=0} (1 - q^(5n+1))(1 - q^(5n+3))(1 - q^(5n+4)).

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 5, 6, 8, 11, 13, 16, 22, 26, 32, 40, 49, 59, 73, 87, 105, 126, 151, 178, 214, 252, 297, 351, 413, 481, 566, 658, 767, 892, 1034, 1195, 1386, 1595, 1838, 2114, 2429, 2781, 3189, 3642, 4160, 4744, 5404, 6141, 6986, 7921, 8980

Offset: 0

Ralf Stephan, May 13 2005

nonn

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

Cf. A035959, A107234, A107235, A107237.

nmax = 50; CoefficientList[Series[1/Product[(1 - x^(5*k+1))*(1 - x^(5*k+3))*(1 - x^(5*k+4)), {k, 0, nmax/5}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 07 2021 *)

a(n) ~ Pi^(3/5) * exp(Pi*sqrt(2*n/5)) / (Gamma(2/5) * 2^(4/5) * 5^(7/10) * n^(4/5)). - Vaclav Kotesovec, Jan 07 2021

A107237 Expansion of 1 / Product_{n>=0} (1 - q^(5n+2))(1 - q^(5n+3))(1 - q^(5n+4)).

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 3, 3, 5, 5, 7, 8, 12, 12, 17, 19, 26, 28, 37, 41, 53, 60, 74, 84, 105, 118, 144, 164, 198, 224, 269, 305, 362, 411, 484, 550, 645, 729, 850, 964, 1117, 1262, 1458, 1647, 1894, 2137, 2446, 2757, 3150, 3542, 4031

Offset: 0

Ralf Stephan, May 13 2005

nonn

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

Cf. A035959, A107234, A107235, A107236.

nmax = 50; CoefficientList[Series[1/Product[(1 - x^(5*k+2))*(1 - x^(5*k+3))*(1 - x^(5*k+4)), {k, 0, nmax/5}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jan 07 2021 *)

a(n) ~ Pi^(4/5) * exp(Pi*sqrt(2*n/5)) / (Gamma(1/5) * 2^(9/10) * 5^(3/5) * n^(9/10)). - Vaclav Kotesovec, Jan 07 2021

A285928 Expansion of (Product_{k>0} (1 - x^(5*k)) / (1 - x^k))^5 in powers of x.

Original entry on oeis.org

1, 5, 20, 65, 190, 501, 1240, 2890, 6440, 13775, 28502, 57205, 111880, 213670, 399620, 733128, 1321850, 2345340, 4100700, 7072520, 12045005, 20272465, 33746060, 55595635, 90706390, 146638756, 235016940, 373580735, 589238640, 922537655, 1434232510, 2214817165

Offset: 0

Seiichi Manyama, Apr 28 2017

nonn

In general, if m > 1 and g.f. = Product_{k>=1} ((1 - x^(m*k)) / (1 - x^k))^m, then a(n, m) ~ exp(Pi*sqrt(2*(m-1)*n/3)) * (m-1)^(1/4) / (2^(5/4) * 3^(1/4) * m^(m/2) * n^(3/4)). - Vaclav Kotesovec, Apr 30 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000

(Product_{k>0} (1 - x^(m*k)) / (1 - x^k))^m: A022567 (m=2), A285927 (m=3), A093160 (m=4), this sequence (m=5).

Cf. A035959, A285932.

nmax = 40; CoefficientList[Series[Product[((1 - x^(5*k)) / (1 - x^k))^5, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 30 2017 *)

a(0) = 1, a(n) = (5/n)*Sum_{k=1..n} A116073(k)*a(n-k) for n > 0.

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(3/4) * 3^(1/4) * 5^(5/2) * n^(3/4)). - Vaclav Kotesovec, Apr 30 2017

cp's OEIS Frontend

A328546 Number of 12-regular partitions of n (no part is a multiple of 12).

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A286653 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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A121591 Expansion of (eta(q^5) / eta(q))^6 in powers of q.

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A145466 Expansion of q^(1/6) * eta(q) / eta(q^5) in powers of q.

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A061199 Upper right triangle read by columns where T(n,k), with k >= n, is the 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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A107234 Expansion of 1 / Product_{n>=0} (1-q^(5n+1))(1-q^(5n+2))(1-q^(5n+3)).

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A107235 Expansion of 1 / Product_{n>=0} (1 - q^(5n+1))*(1 - q^(5n+2))*(1 - q^(5n+4)).

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A107236 Expansion of 1 / Product_{n>=0} (1 - q^(5n+1))*(1 - q^(5n+3))*(1 - q^(5n+4)).

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A107235 Expansion of 1 / Product_{n>=0} (1 - q^(5n+1))(1 - q^(5n+2))(1 - q^(5n+4)).

A107236 Expansion of 1 / Product_{n>=0} (1 - q^(5n+1))(1 - q^(5n+3))(1 - q^(5n+4)).

A107237 Expansion of 1 / Product_{n>=0} (1 - q^(5n+2))(1 - q^(5n+3))(1 - q^(5n+4)).