A035985 -id:A035985 - 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).

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

A320609 Number of parts in all partitions of n in which no part occurs more than six times.

Original entry on oeis.org

1, 3, 6, 12, 20, 35, 47, 78, 111, 165, 228, 330, 444, 614, 820, 1108, 1452, 1930, 2491, 3255, 4175, 5366, 6802, 8665, 10880, 13705, 17089, 21336, 26401, 32716, 40207, 49458, 60452, 73863, 89780, 109101, 131902, 159415, 191864, 230741, 276470, 331021, 394970

Offset: 1

Alois P. Heinz, Oct 17 2018

nonn

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

Column k=6 of A210485.

Cf. A035985.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(3*i*(i+1) [0, l[1]*j]+l)(b(n-i*j, min(n-i*j, i-1))), j=0..min(n/i, 6))))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..50);

Table[Length[Flatten[Select[IntegerPartitions[n], Max[Tally[#][[All, 2]]] <= 6 &]]], {n, 43}] (* Robert Price, Jul 31 2020 *)

a(n) ~ log(7) * exp(2*Pi*sqrt(n/7)) / (2 * Pi * 7^(1/4) * n^(1/4)). - Vaclav Kotesovec, Oct 18 2018

A244600 Expansion of f(-x) / f(-x^7) in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, -1, 0, 0, 1, 0, 2, -1, -1, 0, 0, 0, 0, 3, -3, -2, 0, 0, 1, 0, 5, -3, -3, 0, 0, 2, 0, 8, -6, -5, 0, 0, 3, 0, 11, -8, -7, 0, 0, 3, 0, 17, -13, -11, 0, 0, 6, 0, 24, -17, -14, 0, 0, 7, 0, 34, -25, -21, 0, 0, 11, 0, 47, -33, -28, 0, 0, 14, 0, 64, -47, -39

Offset: 0

Michael Somos, Jul 01 2014

sign

			G.f. = 1 - x - x^2 + x^5 + 2*x^7 - x^8 - x^9 + 3*x^14 - 3*x^15 - 2*x^16 + ...
G.f. = q^-1 - q^3 - q^7 + q^19 + 2*q^27 - q^31 - q^35 + 3*q^55 - 3*q^59 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A035985.

a[ n_] := SeriesCoefficient[ QPochhammer[ x] / QPochhammer[ x^7], {x, 0, n}];

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^7 + A), n))};

Expansion of q^(1/4) * eta(q) / eta(q^7) in powers of q.
Euler transform of period 7 sequence [ -1, -1, -1, -1, -1, -1, 0, ...].
Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u - v^3) * (u^3 - v) - 3*u*v * (2 + u*v).
G.f. is a period 1 Fourier series which satisfies f(-1 / (112 t)) = 7^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A035985.
G.f.: Product_{k>0} (1 - x^k) / (1 - x^(7*k)).
Convolution inverse of A035985.
a(7*n + 3) = a(7*n + 4) = a(7*n + 6) = 0.
a(n) = -(1/n)*Sum_{k=1..n} A113957(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017

A124094 Table T(n,m) giving number of partitions of n such that all parts are coprime to m. Read along antidiagonals (increasing n, decreasing m).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 5, 1, 1, 1, 2, 2, 7, 1, 1, 2, 2, 4, 3, 11, 1, 1, 1, 3, 2, 5, 4, 15, 1, 1, 2, 1, 5, 3, 7, 5, 22, 1, 1, 1, 3, 1, 6, 4, 9, 6, 30, 1, 1, 2, 2, 5, 2, 10, 5, 13, 8, 42, 1, 1, 1, 2, 2, 7, 2, 13, 6, 16, 10, 56, 1, 1, 2, 2, 4, 3, 11, 3, 19, 8, 22, 12, 77, 1, 1, 1, 3, 2, 5, 4

Offset: 0

R. J. Mathar, Nov 26 2006

			Upper left corner of table starts with row m=1 and column n=0:
1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,490,627,792,1002,1255,
1,1,1,2,2,3, 4, 5, 6, 8,10,12,15, 18, 22, 27, 32, 38, 46, 54, 64, 76,  89, 104,
1,1,2,2,4,5, 7, 9,13,16,22,27,36, 44, 57, 70, 89,108,135,163,202,243, 297, 355,
1,1,1,2,2,3, 4, 5, 6, 8,10,12,15, 18, 22, 27, 32, 38, 46, 54, 64, 76,  89, 104,
1,1,2,3,5,6,10,13,19,25,34,44,60, 76,100,127,164,205,262,325,409,505, 628, 769,
1,1,1,1,1,2, 2, 3, 3, 3, 4, 5, 6,  7,  8,  9, 10, 12, 14, 16, 18, 20,  23,  26,
1,1,2,3,5,7,11,14,21,28,39,51,70, 90,119,153,199,252,324,406,515,642, 804, 994,
1,1,1,2,2,3, 4, 5, 6, 8,10,12,15, 18, 22, 27, 32, 38, 46, 54, 64, 76,  89, 104,
1,1,2,2,4,5, 7, 9,13,16,22,27,36, 44, 57, 70, 89,108,135,163,202,243, 297, 355,
1,1,1,2,2,2, 3, 4, 4, 6, 7, 8,10, 12, 14, 16, 19, 22, 26, 30, 35, 41,  47,  54,
1,1,2,3,5,7,11,15,22,30,42,55,76, 99,132,171,224,286,370,468,597,750, 945,1177,
1,1,1,1,1,2, 2, 3, 3, 3, 4, 5, 6,  7,  8,  9, 10, 12, 14, 16, 18, 20,  23,  26,
1,1,2,3,5,7,11,15,22,30,42,56,77,100,134,174,228,292,378,479,612,770, 972,1213,
1,1,1,2,2,3, 4, 4, 5, 7, 8,10,12, 14, 17, 21, 24, 28, 34, 39, 46, 53,  61,  71,
1,1,2,2,4,4, 6, 7,11,12,16,19,25, 29, 37, 44, 56, 65, 80, 94,114,133, 160, 187,
1,1,1,2,2,3, 4, 5, 6, 8,10,12,15, 18, 22, 27, 32, 38, 46, 54, 64, 76,  89, 104,
1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,296,384,488,624,787, 995,1244,
1,1,1,1,1,2, 2, 3, 3, 3, 4, 5, 6,  7,  8,  9, 10, 12, 14, 16, 18, 20,  23,  26,
1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,489,626,790, 999,1250,
1,1,1,2,2,2, 3, 4, 4, 6, 7, 8,10, 12, 14, 16, 19, 22, 26, 30, 35, 41,  47,  54,

Alois P. Heinz, Antidiagonals n = 0..140, flattened
N. Robbins, On partition functions and divisor sums, J. Int. Sequences, 5 (2002) 02.1.4.

Row m=1 is A000041. Rows m=2, 4, 8, ... (where m is a power of 2) are A000009. Rows m=3, 9, ... (where m is a power of 3) are A000726. Row m=5 is A035959. Row=7 is A035985. Row m=10 is A096938.

b:= proc(n, i, m) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i<1 then 0
    else b(n, i-1, m) +`if`(igcd(m, i)=1, b(n-i, i, m), 0)
      fi
    end:
T:= (n, m)-> b(n, n, m):
seq (seq (T(n, 1+d-n), n=0..d), d=0..13);  # Alois P. Heinz, Sep 28 2011

b[n_, i_, m_] := b[n, i, m] = Which[n < 0, 0, n == 0, 1, i < 1, 0, True, b[n, i-1, m] + If[GCD[m, i] == 1, b[n-i, i, m], 0]]; t[n_, m_] := b[n, n, m]; Table[Table[t[n, 1+d-n], {n, 0, d}], {d, 0, 13}] // Flatten (* Jean-François Alcover, Jan 10 2014, translated from Alois P. Heinz's Maple code *)

sigmastar(n,m)= { local(d,res=0) ; d=divisors(n) ; for(i=1,matsize(d)[2], if( gcd(d[i],m)==1, res += d[i] ; ) ; ) ; return(res) ; } f(n,m)= { local(qvec=vector(n+1,i,gcd(1,m))) ; for(i=1,n, qvec[i+1]=sum(k=1,i,sigmastar(k,m)*qvec[i-k+1])/i ; ) ; return(qvec[n+1]) ; } { for(d=1,18, for(c=0,d-1, r=d-c ; print1(f(c,r),",") ; ) ; ) ; }

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A320609 Number of parts in all partitions of n in which no part occurs more than six times.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A244600 Expansion of f(-x) / f(-x^7) in powers of x where f() is a Ramanujan theta function.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A124094 Table T(n,m) giving number of partitions of n such that all parts are coprime to m. Read along antidiagonals (increasing n, decreasing m).

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI