cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A035939 Number of partitions of n into parts not of the form 7k, 7k+3 or 7k-3. Also number of partitions such that the differences between parts at distance 2 are greater than 1.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 6, 7, 10, 12, 16, 19, 25, 30, 38, 46, 57, 68, 84, 99, 121, 143, 172, 202, 242, 283, 336, 392, 462, 537, 630, 729, 851, 982, 1140, 1312, 1518, 1741, 2006, 2295, 2635, 3007, 3442, 3917, 4470, 5077, 5776, 6545, 7429, 8399, 9510, 10731
Offset: 0

Views

Author

Olivier Gérard

Keywords

Comments

Case k=3, i=3 of Gordon Theorem.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 7*x^7 + 10*x^8 + ...
G.f. = q^-1 + q^41 + 2*q^83 + 2*q^125 + 3*q^167 + 4*q^209 + 6*q^251 + ...

References

  • G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.

Links

Crossrefs

Cf. A035937, A035938.

Programs

  • Maple
    # See A035937 for GordonsTheorem
A035939_list := n -> GordonsTheorem([1, 1, 0, 0, 1, 1, 0], n):
A035939_list(40); # Peter Luschny, Jan 22 2012
  • Mathematica
    a[ n_] := SeriesCoefficient[ 1 / Product[ (1 - x^(7 k - 1)) (1 - x^(7 k - 2)) (1 - x^(7 k - 5)) (1 - x^(7 k - 6)), {k, Ceiling[n/7]}], {x, 0, n}]; (* Michael Somos, Dec 29 2014 *)
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x^1, x^7] QPochhammer[ x^2, x^7] QPochhammer[ x^5, x^7] QPochhammer[ x^6, x^7] ), {x, 0, n}]; (* Michael Somos, Dec 29 2014 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 -[0, 1, 1, 0, 0, 1, 1][k%7 + 1] * x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Dec 29 2014 */
  • Sage
    # See A035937 for GordonsTheorem
def A035939_list(len) :  return GordonsTheorem([1, 1, 0, 0, 1, 1, 0], len)
A035939_list(40) # Peter Luschny, Jan 22 2012

Formula

Expansion of f(-x^3, -x^4) / f(-x, -x^2) in powers of x where f() is Ramanujan's two-variable theta function. - Michael Somos, Dec 29 2014
Euler transform of period 7 sequence [ 1, 1, 0, 0, 1, 1, 0, ...].- Michael Somos, Dec 29 2014
G.f.: 1 / (Product_{k>0} (1 - x^(7*k - 6)) * (1 - x^(7*k - 5)) * (1 - x^(7*k - 2)) * (1 - x^(7*k - 1))). - Michael Somos, Dec 29 2014
G.f.: (Product_{k>0} (1 + x^k)) * (Sum_{k>=0} x^(2*k^2) / (Product_{i=1..k} (1 - x^(2*i)) * (1 + x^(2*i-1)) * (1 + x^(2*i)))). - Michael Somos, Dec 31 2014
a(n) ~ 2^(1/4) * cos(Pi/14) * exp(2*Pi*sqrt(2*n/21)) / (3^(1/4) * 7^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 13 2015

Extensions

Missing a(0)=1 prepended by Michael Somos, Dec 29 2014
Name simplified by George Beck, Aug 27 2023

A035938 Number of partitions in parts not of the form 7k, 7k+2 or 7k-2. Also number of partitions with 1 part of size 1 and differences between parts at distance 2 are greater than 1.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 5, 6, 8, 10, 13, 16, 21, 25, 31, 38, 47, 56, 69, 82, 99, 118, 141, 166, 199, 233, 275, 322, 379, 440, 516, 598, 696, 805, 933, 1074, 1242, 1425, 1639, 1878, 2154, 2458, 2812, 3202, 3650, 4148, 4716, 5344, 6064, 6857, 7758, 8758, 9888, 11136
Offset: 0

Views

Author

Olivier Gérard

Keywords

Comments

Case k=3,i=2 of Gordon Theorem.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 5*x^6 + 6*x^7 + 8*x^8 + ...
G.f. = q^5 + q^47 + q^89 + 2*q^131 + 3*q^173 + 3*q^215 + 5*q^257 + ...

References

  • G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.

Links

Crossrefs

Cf. A035937, A035939.

Programs

  • Maple
    with(numtheory): a:= proc(n) option remember; `if`(n=0, 1, add(add(d*[0, 1, 0, 1, 1, 0, 1][1+irem(d, 7)], d=divisors(j)) *a(n-j), j=1..n)/n) end: seq(a(n), n=1..100); # Alois P. Heinz, Jan 22 2012
  • Mathematica
    a[n_] := a[n] = If[n == 0, 1, Sum[Sum[d*{0, 1, 0, 1, 1, 0, 1}[[1+Mod[d, 7]]], {d, Divisors[j]}]*a[n-j], {j, 1, n}]/n]; Table[a[n], {n, 0, 52}] (* Jean-François Alcover, Feb 25 2014, after Alois P. Heinz *)
a[ n_] := SeriesCoefficient[ 1 / Product[ (1 - x^(7 k - 1)) (1 - x^(7 k - 3)) (1 - x^(7 k - 4)) (1 - x^(7 k - 6)), {k, Ceiling[n/7]}], {x, 0, n}]; (* Michael Somos, Dec 30 2014 *)
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x^1, x^7] QPochhammer[ x^3, x^7] QPochhammer[ x^4, x^7] QPochhammer[ x^6, x^7] ), {x, 0, n}]; (* Michael Somos, Dec 30 2014 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 - [0, 1, 0, 1, 1, 0, 1][k%7 + 1] * x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Feb 03 2012 */
  • Sage
    # See A035937 for GordonsTheorem
def A035938_list(len) :  return GordonsTheorem([1, 0, 1, 1, 0, 1, 0], len)
A035938_list(40) # Peter Luschny, Jan 22 2012

Formula

Expansion of f(-x^2, -x^5) / f(-x, -x^2) in powers of x where f() is Ramanujan's two-variable theta function. - Michael Somos, Dec 30 2014
Euler transform of period 7 sequence [ 1, 0, 1, 1, 0, 1, 0, ...]. - Michael Somos, Feb 03 2012
G.f.: 1 / (Product_{k>0} (1 - x^(7*k - 6)) * (1 - x^(7*k - 4)) * (1 - x^(7*k - 3)) * (1 - x^(7*k - 1))). - Michael Somos, Feb 03 2012
G.f.: (Product_{k>0} (1 + x^k)) * (Sum_{k>0} x^(2*k + 2*k^2) / (Product_{i=1..k} (1 - x^(2*i)) * (1 + x^(2*i-1)) * (1 + x^(2*i)))).
a(n) ~ 2^(1/4) * cos(3*Pi/14) * exp(2*Pi*sqrt(2*n/21)) / (3^(1/4) * 7^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 13 2015

A035940 Number of partitions in parts not of the form 9k, 9k+1 or 9k-1. Also number of partitions with no part of size 1 and differences between parts at distance 3 are greater than 1.

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 4, 6, 7, 10, 12, 17, 19, 26, 31, 40, 47, 61, 71, 90, 106, 131, 154, 190, 222, 270, 317, 381, 445, 533, 620, 737, 857, 1011, 1173, 1379, 1593, 1863, 2151, 2503, 2881, 3343, 3837, 4435, 5083, 5853, 6693, 7688, 8769, 10043, 11437, 13061
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

Case k=4, i=1 of Gordon Theorem.

References

  • G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.

Programs

  • Maple
    # See A035937 for GordonsTheorem
A035940_list := n -> GordonsTheorem([0, 1, 1, 1, 1, 1, 1, 0, 0], n):
A035940_list(40) # Peter Luschny, Jan 22 2012
  • Mathematica
    nmax = 60; Rest[CoefficientList[Series[Product[1 / ((1 - x^(9*k-2)) * (1 - x^(9*k-3)) * (1 - x^(9*k-4)) * (1 - x^(9*k-5)) * (1 - x^(9*k-6)) * (1 - x^(9*k-7)) ), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Nov 12 2015 *)
  • Sage
    # See A035937 for GordonsTheorem
def A035940_list(len) :  return GordonsTheorem([0, 1, 1, 1, 1, 1, 1, 0, 0], len)
A035940_list(40) # Peter Luschny, Jan 22 2012

Formula

a(n) ~ exp(2*Pi*sqrt(n)/3) / (6 * (1+2*cos(2*Pi/9)) * n^(3/4)). - Vaclav Kotesovec, Nov 12 2015

A035941 Number of partitions of n into parts not of the form 9k, 9k+2 or 9k-2. Also number of partitions with 1 part of size 1 and differences between parts at distance 3 are greater than 1.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 7, 10, 13, 17, 21, 28, 35, 44, 55, 69, 84, 105, 127, 156, 189, 229, 275, 333, 397, 475, 565, 673, 795, 943, 1109, 1307, 1533, 1798, 2099, 2455, 2855, 3323, 3855, 4472, 5169, 5978, 6890, 7942, 9132, 10495, 12032, 13796, 15778, 18040
Offset: 1

Views

Author

Olivier Gérard

Keywords

Comments

Case k=4, i=2 of Gordon Theorem.

References

  • G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.

Programs

  • Maple
    # See A035937 for GordonsTheorem
A035941_list := n -> GordonsTheorem([1, 0, 1, 1, 1, 1, 0, 1, 0], n):
A035941_list(40); # Peter Luschny, Jan 22 2012
  • Mathematica
    nmax = 60; Rest[CoefficientList[Series[Product[1 / ((1 - x^(9*k-1)) * (1 - x^(9*k-3)) * (1 - x^(9*k-4)) * (1 - x^(9*k-5)) * (1 - x^(9*k-6)) * (1 - x^(9*k-8)) ), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Nov 12 2015 *)
  • Sage
    # See A035937 for GordonsTheorem
def A035941_list(len) :  return GordonsTheorem([1, 0, 1, 1, 1, 1, 0, 1, 0], len)
A035941_list(40) # Peter Luschny, Jan 22 2012

Formula

a(n) ~ sin(2*Pi/9) * exp(2*Pi*sqrt(n)/3) / (3*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 12 2015
Showing 1-4 of 4 results.

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