A035959 -id:A035959 - cp's OEIS frontend

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).

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

A296163 a(n) = [x^n] Product_{k>=1} ((1 - x^(5*k))/(1 - x^k))^n.

Original entry on oeis.org

1, 1, 5, 22, 105, 501, 2456, 12160, 60801, 306130, 1550255, 7887034, 40281720, 206405967, 1060602800, 5463059772, 28199365873, 145832364580, 755420838614, 3918935839970, 20357605331355, 105878815699042, 551273881133750, 2873161931172668, 14988243880188600

Offset: 0

Ilya Gutkovskiy, Dec 06 2017

nonn

Eric Weisstein's World of Mathematics, Partition Function b_k

Cf. A008485, A035959, A121591, A263002, A270913, A285928, A296044, A296162.

Table[SeriesCoefficient[Product[((1 - x^(5 k))/(1 - x^k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 24}]
Table[SeriesCoefficient[Product[(1 + x^k + x^(2 k) + x^(3 k) + x^(4 k))^n, {k, 1, n}], {x, 0, n}], {n, 0, 24}]
(* Calculation of constant d: *) With[{k = 5}, 1/r /. FindRoot[{s == QPochhammer[(r*s)^k] / QPochhammer[r*s], k*(-(s*QPochhammer[r*s]*(Log[1 - (r*s)^k] + QPolyGamma[0, 1, (r*s)^k]) / Log[(r*s)^k]) + (r*s)^k * Derivative[0, 1][QPochhammer][(r*s)^k, (r*s)^k]) == s*QPochhammer[r*s] + s^2*(-(QPochhammer[r*s]*(Log[1 - r*s] + QPolyGamma[0, 1, r*s]) / (s*Log[r*s])) + r*Derivative[0, 1][QPochhammer][r*s, r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 70]] (* Vaclav Kotesovec, Jan 17 2024 *)

a(n) = [x^n] Product_{k>=1} (1 + x^k + x^(2*k) + x^(3*k) + x^(4*k))^n.

a(n) ~ c * d^n / sqrt(n), where d = 5.3271035802753567624196808294779171420899175782347488197... and c = 0.2712048688090020853684153670711011713396954... - Vaclav Kotesovec, May 13 2018

A213598 Number of partitions of n in which no parts are multiples of 49.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, 231, 297, 385, 490, 627, 792, 1002, 1255, 1575, 1958, 2436, 3010, 3718, 4565, 5604, 6842, 8349, 10143, 12310, 14883, 17977, 21637, 26015, 31185, 37338, 44583, 53174, 63261, 75175, 89134, 105558, 124754, 147273, 173524

Offset: 0

Michael Somos, Jun 14 2012

nonn

For n<49 we have a(n)=A000041(n), for n>=49 a(n)!=A000041(n).

In Fricke page 401, he gives the expansion sigma(omega) = q^4 + q^6 + 2q^8 + 3q^10 + 5q^12 + 7q^14 + 11q^16 + 15q^18 + ... where q = exp( Pi i omega).

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 7*x^5 + 11*x^6 + 15*x^7 + 22*x^8 + ...
G.f. = q^2 + q^3 + 2*q^4 + 3*q^5 + 5*q^6 + 7*q^7 + 11*q^8 + 15*q^9 + 22*q^10 + ...

R. Fricke, Die elliptischen Funktionen und ihre Anwendungen, Teubner, 1922, Vol. 2, see p. 401. Eq. (49)

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

Cf. A000009 (m=2), A000726 (m=3), A001935 (m=4), A035959 (m=5), A219601 (m=6), A035985 (m=7), A261775 (m=8), A104502 (m=9), A261776 (m=10), A092885 (m=25), this sequence (m=49).

Cf. A000041, A287926.

a[ n_] := SeriesCoefficient[ Product[ 1 - x^k, {k, 49, n, 49}] / Product[ 1 - x^k, {k, n}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ x^49] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, May 13 2014 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^49 + A) / eta(x + A), n))};

Expansion of q^(-2) * eta(q^49) / eta(q) in powers of q.
Euler transform of period 49 sequence [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, ...].
Given g.f. A(x) then B(x) = x^2 * A(x) satisfies 0 = f(B(x), B(x^2),
B(x^4)) where f(u, v, w) = u * v * w * (1 - 7*v^2) - (v - w) * (u - v) * (v^2 - u*w).
G.f. is a period 1 Fourier series which satisfies f(-1 / (49 t)) = 1 / (7 f(t)) where q = exp(2 Pi i t).
G.f.: Product_{k>0} (1 - x^(49*k)) / (1 - x^k).
a(n) ~ exp(4*Pi*sqrt(2*n)/7) / (2^(1/4) * 7^(3/2) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015
a(n) = (1/n)*Sum_{k=1..n} A287926(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Jun 16 2017

A320607 Number of parts in all partitions of n in which no part occurs more than four times.

Original entry on oeis.org

1, 3, 6, 12, 15, 29, 41, 65, 91, 132, 179, 257, 339, 465, 616, 823, 1062, 1402, 1790, 2320, 2939, 3750, 4701, 5946, 7398, 9243, 11428, 14161, 17368, 21372, 26056, 31823, 38596, 46838, 56499, 68208, 81868, 98292, 117489, 140390, 167068, 198796, 235655, 279239

Offset: 1

Alois P. Heinz, Oct 17 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..5000

Column k=4 of A210485.

Cf. A035959.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(2*i*(i+1) [0, l[1]*j]+l)(b(n-i*j, min(n-i*j, i-1))), j=0..min(n/i, 4))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..50);

Table[Length[Flatten[Select[IntegerPartitions[n],Max[Tally[#][[All,2]]]<5&]]],{n,50}] (* Harvey P. Dale, May 18 2020 *)

a(n) ~ 3^(1/4) * log(5) * exp(2*Pi*sqrt(2*n/15)) / (2^(5/4) * 5^(1/4) * Pi * n^(1/4)). - Vaclav Kotesovec, Oct 18 2018

A332311 Number of compositions (ordered partitions) of n into distinct parts where no part is a multiple of 5.

Original entry on oeis.org

1, 1, 1, 3, 3, 4, 9, 11, 11, 19, 44, 31, 61, 87, 117, 144, 279, 311, 389, 541, 640, 1003, 1225, 2145, 2493, 3452, 3507, 5417, 6671, 8821, 11580, 17959, 21043, 26289, 34797, 41536, 59637, 72707, 85871, 110947, 172472, 175873, 249691, 327801, 418779, 512748

Offset: 0

Ilya Gutkovskiy, Feb 09 2020

nonn

			a(6) = 9 because we have [6], [4, 2], [3, 2, 1], [3, 1, 2], [2, 4], [2, 3, 1], [2, 1, 3], [1, 3, 2] and [1, 2, 3].

Alois P. Heinz, Table of n, a(n) for n = 0..5000
Index entries for sequences related to compositions

Cf. A032020, A032021, A035959, A096938, A332309, A332310.

b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$2, 0):
seq(a(n), n=0..55);  # Alois P. Heinz, Feb 09 2020

b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0, If[n == 0, p!, Sum[b[n - i j, i - 1, p + j], {j, 0, Min[Mod[i, 5], 1, n/i]}]]];
a[n_] := b[n, n, 0];
a /@ Range[0, 55] (* Jean-François Alcover, Nov 17 2020, after Alois P. Heinz *)

A259357 Expansion of f(-x^5)^2 / f(-x, -x^4) in powers of x where f(,) is the Ramanujan general theta function.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 3, 3, 3, 3, 5, 5, 5, 6, 7, 7, 9, 9, 11, 11, 13, 13, 16, 17, 19, 20, 23, 24, 27, 29, 32, 34, 38, 40, 46, 48, 52, 56, 62, 65, 72, 76, 84, 89, 97, 102, 113, 119, 129, 137, 149, 157, 171, 181, 196, 208, 224, 236, 256, 270, 290, 308, 331

Offset: 0

Michael Somos, Jun 24 2015

nonn

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

Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).

			G.f. = 1 + x + x^2 + x^3 + 2*x^4 + x^5 + 2*x^6 + 2*x^7 + 3*x^8 + 3*x^9 + ...
G.f. = q^23 + q^143 + q^263 + q^383 + 2*q^503 + q^623 + 2*q^743 + 2*q^863 + ...

G. E. Andrews and B. C. Berndt, Ramanujan's Lost Notebook, Part III, Springer, 2012, see p. 12, Entry 2.1.3, Equation (2.1.21).
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, equation 3.

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A035959, A113428, A259358.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^5] / (QPochhammer[ x, x^5] QPochhammer[ x^4, x^5]), {x, 0, n}];

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, -1, 0, 0, -1][k%5+1]), n))};

Expansion of f(-x^5) * f(-x^2, -x^3) / f(-x) in powers of x where f(,) is the Ramanujan general theta function.
Expansion of f(-x^5) * G(x) in powers of x where f() is a Ramanujan theta function and G() is a Rogers-Ramanujan function. - Michael Somos, Jul 09 2015
Euler transform of period 5 sequence [ 1, 0, 0, 1, -1, ...].
G.f.: Product_{k>0} (1 - x^(5*k)) / ((1 - x^(5*k-4)) * (1 - x^(5*k-1))).
Convolution of A035959 and A113428.

A259358 Expansion of f(-x^5)^2 / f(-x^2, -x^3) in powers of x where f(,) is the Ramanujan general theta function.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 2, 1, 2, 2, 2, 2, 3, 2, 4, 4, 5, 4, 6, 5, 7, 7, 8, 8, 11, 10, 12, 13, 15, 14, 18, 17, 21, 21, 24, 25, 29, 29, 34, 35, 39, 40, 47, 47, 53, 55, 61, 63, 72, 73, 82, 86, 94, 97, 109, 112, 124, 129, 141, 147, 162, 167, 183, 192, 208, 217, 237

Offset: 0

Michael Somos, Jun 24 2015

nonn

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

Rogers-Ramanujan functions: G(q) (see A003114), H(q) (A003106).

			G.f. = 1 + x^2 + x^3 + x^4 + 2*x^6 + x^7 + 2*x^8 + 2*x^9 + 2*x^10 + ...
G.f. = q^47 + q^287 + q^407 + q^527 + 2*q^767 + q^887 + 2*q^1007 + ...

G. E. Andrews and B. C. Berndt, Ramanujan's Lost Notebook, Part III, Springer, 2012, see p. 12, Entry 2.1.3, Equation (2.1.22).
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, equation 4.

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A035959, A113429, A259357.

a[ n_] := SeriesCoefficient[ QPochhammer[ x^5] / (QPochhammer[ x^2, x^5] QPochhammer[ x^3, x^5]), {x, 0, n}];

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, 0, -1, -1, 0][k%5+1]), n))};

Expansion of f(-x^5) * f(-x, -x^4) / f(-x) in powers of x where f(,) is the Ramanujan general theta function.
Expansion of f(-x^5) * H(x) in powers of x where f() is a Ramanujan theta funcation and H() is a Rogers-Ramanujan function. - Michael Somos, Jul 09 2015
Euler transform of period 5 sequence [ 0, 1, 1, 0, -1, ...].
G.f.: Product_{k>0} (1 - x^(5*k)) / ((1 - x^(5*k-3)) * (1 - x^(5*k-2))).
Convolution of A035959 and A113429.

A262364 Expansion of Product_{k>=1} (1-x^(5k))/(1-x^(2k)).

Original entry on oeis.org

1, 0, 1, 0, 2, -1, 3, -1, 5, -2, 6, -3, 10, -5, 13, -7, 19, -11, 25, -15, 35, -22, 45, -30, 62, -41, 79, -55, 105, -75, 134, -98, 175, -130, 220, -168, 284, -219, 355, -280, 451, -360, 561, -455, 705, -578, 870, -725, 1085, -910, 1331, -1132, 1644, -1410

Offset: 0

Vaclav Kotesovec, Sep 23 2015

sign

In general, if m >= 1 and g.f. = Product_{k>=1} (1-x^((2*m+1)*k))/(1-x^(2*k)), then a(n) ~ (-1)^n * exp(Pi*sqrt((4*m+1)*n/(6*(2*m+1)))) * (4*m+1)^(1/4) / (2^(7/4) * 3^(1/4) * (2*m+1)^(3/4) * n^(3/4)).

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

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016, p. 15.

Cf. A035959, A147783, A262346.

nmax = 60; CoefficientList[Series[Product[(1-x^(5*k))/(1-x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x]
CoefficientList[Series[QPochhammer[x^5]/QPochhammer[x^2], {x, 0, 60}], x]

lista(nn) = {q='q+O('q^nn); Vec(eta(q^5)/eta(q^2))} \\ Altug Alkan, Mar 21 2018

a(n) ~ (-1)^n * 3^(1/4) * exp(Pi*sqrt(3*n/10)) / (2^(7/4) * 5^(3/4) * n^(3/4)).

A278668 Expansion of Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^3 in powers of x.

Original entry on oeis.org

1, 3, 9, 22, 51, 107, 218, 420, 788, 1428, 2531, 4375, 7430, 12377, 20313, 32833, 52402, 82585, 128750, 198588, 303428, 459375, 689710, 1027243, 1518709, 2229375, 3251022, 4710777, 6785378, 9717677, 13841991, 19614182, 27656250, 38810312, 54216128, 75406438

Offset: 0

Seiichi Manyama, Nov 25 2016

nonn

			G.f.: 1 + 3*x + 9*x^2 + 22*x^3 + 51*x^4 + 107*x^5 + 218*x^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^k: A035959 (k=1), A160461 (k=2), this sequence (k=3), A278680 (k=4), A277212 (k=5), A182821 (k=6).

nmax = 30; CoefficientList[Series[Product[(1 - x^(5*k))/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)

G.f.: Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^3.

a(n) ~ exp(2*Pi*sqrt(7*n/15)) * 7^(3/4) / (20 * 3^(3/4) * 5^(1/4) * n^(5/4)). - Vaclav Kotesovec, Nov 10 2017

A278680 Expansion of Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^4 in powers of x.

Original entry on oeis.org

1, 4, 14, 40, 105, 251, 570, 1226, 2540, 5075, 9855, 18630, 34439, 62340, 110805, 193624, 333235, 565415, 947040, 1567130, 2564425, 4152535, 6658711, 10579380, 16663755, 26033200, 40357641, 62106290, 94912385, 144088840, 217368655, 325945320, 485950150, 720515475

Offset: 0

Seiichi Manyama, Nov 25 2016

nonn

			G.f.: 1 + 4*x + 14*x^2 + 40*x^3 + 105*x^4 + 251*x^5 + 570*x^6 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^k: A035959 (k=1), A160461 (k=2), A278668 (k=3), this sequence (k=4), A277212 (k=5), A182821 (k=6).

nmax = 30; CoefficientList[Series[Product[(1 - x^(5*k))/(1 - x^k)^4, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2017 *)

G.f.: Product_{n>=1} (1 - x^(5*n))/(1 - x^n)^4.

a(n) ~ 19 * exp(Pi*sqrt(38*n/15)) / (120 * sqrt(10) * n^(3/2)). - Vaclav Kotesovec, Nov 10 2017

cp's OEIS Frontend

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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A296163 a(n) = [x^n] Product_{k>=1} ((1 - x^(5*k))/(1 - x^k))^n.

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A213598 Number of partitions of n in which no parts are multiples of 49.

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A320607 Number of parts in all partitions of n in which no part occurs more than four times.

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A332311 Number of compositions (ordered partitions) of n into distinct parts where no part is a multiple of 5.

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A259357 Expansion of f(-x^5)^2 / f(-x, -x^4) in powers of x where f(,) is the Ramanujan general theta function.

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A259358 Expansion of f(-x^5)^2 / f(-x^2, -x^3) in powers of x where f(,) is the Ramanujan general theta function.

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A262364 Expansion of Product_{k>=1} (1-x^(5*k))/(1-x^(2*k)).

A262364 Expansion of Product_{k>=1} (1-x^(5k))/(1-x^(2k)).