A219601 -id:A219601 - cp's OEIS frontend

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).

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).

A328548 Number of 6-regular bipartitions of n.

Original entry on oeis.org

1, 2, 5, 10, 20, 36, 63, 106, 175, 280, 441, 680, 1034, 1548, 2290, 3346, 4840, 6930, 9837, 13844, 19337, 26810, 36925, 50530, 68741, 92984, 125113, 167490, 223155, 295960, 390825, 513954, 673214, 878480, 1142190, 1479892, 1911051, 2459896, 3156602

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.

Number of r-regular bipartitions of n for r = 2,3,4,5,6: A022567, A328547, A001936, A263002, A328548.

Cf. A219601.

f:=(k,M) -> mul(1-q^(k*j),j=1..M);
LRBP := (L,M) -> (f(L,M)/f(1,M))^2;
S := L -> seriestolist(series(LRBP(L,80),q,60));
S(6);

nmax = 40; CoefficientList[Series[Product[(1 - x^(6*k))/(1 - x^k), {k, 1, nmax}]^2, {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 08 2024 *)

a(n) ~ 5^(1/4) * exp(Pi*sqrt(10*n)/3) / (2^(9/4) * 3^(3/2) * n^(3/4)). - Vaclav Kotesovec, Oct 08 2024

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

A320608 Number of parts in all partitions of n in which no part occurs more than five times.

Original entry on oeis.org

1, 3, 6, 12, 20, 29, 47, 71, 104, 150, 213, 292, 405, 547, 736, 977, 1292, 1688, 2198, 2834, 3638, 4636, 5890, 7431, 9347, 11684, 14564, 18067, 22348, 27531, 33813, 41378, 50504, 61438, 74561, 90208, 108896, 131096, 157485, 188717, 225682, 269285, 320691

Offset: 1

Alois P. Heinz, Oct 17 2018

nonn

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

Column k=5 of A210485.

Cf. A219601.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(5*i*(i+1)/2 [0, l[1]*j]+l)(b(n-i*j, min(n-i*j, i-1))), j=0..min(n/i, 5))))
    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, 43}] (* Robert Price, Jul 31 2020 *)

a(n) ~ log(6) * exp(Pi*sqrt(5*n)/3) / (2 * Pi * 5^(1/4) * n^(1/4)). - Vaclav Kotesovec, Oct 18 2018

A280874 Expansion of Product_{k>=1} (1 - x^(6k)) (1 + x^k) / (1 - x^k).

Original entry on oeis.org

1, 2, 4, 8, 14, 24, 39, 62, 96, 146, 218, 320, 463, 662, 936, 1310, 1816, 2496, 3404, 4608, 6196, 8278, 10994, 14520, 19076, 24938, 32448, 42032, 54218, 69656, 89149, 113680, 144456, 182952, 230966, 290688, 364774, 456446, 569600, 708938, 880128, 1089984

Offset: 0

Vaclav Kotesovec, Jan 09 2017

nonn

Convolution of A219601 and A000009.

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
Andrew Sills, Towards an Automation of the Circle Method, Gems in Experimental Mathematics in Contemporary Mathematics, 2010.

Cf. A000009, A219601.

nmax = 60; CoefficientList[Series[Product[(1-x^(6*k))*(1+x^k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ Pi*sqrt(2) * BesselI(1, sqrt(8*n+2)*Pi/3) / (3*sqrt(12*n+3)).

a(n) ~ exp(2*Pi*sqrt(2*n)/3) / (6*2^(3/4)*n^(3/4)) * (1 + (Pi/6 - 9/(16*Pi))/sqrt(2*n) + (Pi^2/144 - 135/(1024*Pi^2) - 15/64)/n).

A336766 The number of partitions of n into an even number of parts, each part occurring at most five times, minus the number of partitions of n into an odd number of parts, each part occurring at most five times.

Original entry on oeis.org

1, -1, 0, -1, 1, -1, 0, 0, 2, -1, 1, -1, 1, -1, 1, -1, 2, -2, 1, -2, 2, -2, 1, -2, 3, -3, 2, -2, 3, -3, 3, -3, 4, -4, 3, -4, 5, -4, 4, -4, 6, -5, 5, -6, 6, -7, 6, -6, 8, -8, 7, -8, 9, -9, 8, -9, 11, -11, 10, -11, 12, -12, 11, -13, 15, -15, 14, -15, 17, -17, 16, -17

Offset: 0

Jeremy Lovejoy, Aug 04 2020

sign

			There are 10 partitions of 6 where parts occur at most five times: 6, 5+1, 4+2, 4+1+1, 3+3, 3+2+1, 3+1+1+1, 2+2+2, 2+2+1+1, 2+1+1+1+1, and so a(6) = 0.

H. L. Alder and A. A. Muwafi, Identities relating the number of partitions into an even and odd number of parts, Fibonacci Quarterly, 13 (1975), 147-149.

Cf. A000041, A106459, A219601, A336767.

G.f.: Product_{n>0} ((1-q^(6*n))/(1+q^n)).

cp's OEIS Frontend

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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A328548 Number of 6-regular bipartitions of n.

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

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A320608 Number of parts in all partitions of n in which no part occurs more than five times.

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A280874 Expansion of Product_{k>=1} (1 - x^(6*k)) * (1 + x^k) / (1 - x^k).

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A336766 The number of partitions of n into an even number of parts, each part occurring at most five times, minus the number of partitions of n into an odd number of parts, each part occurring at most five times.

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Formula

A280874 Expansion of Product_{k>=1} (1 - x^(6k)) (1 + x^k) / (1 - x^k).