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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 1-3 of 3 results.

A035937 Number of partitions in parts not of the form 7k, 7k+1 or 7k-1. Also number of partitions with no part of size 1 and differences between parts at distance 2 are greater than 1.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 3, 3, 5, 6, 8, 9, 13, 14, 19, 22, 28, 32, 41, 47, 59, 68, 83, 96, 117, 134, 161, 186, 221, 254, 301, 344, 405, 464, 541, 619, 720, 820, 949, 1081, 1245, 1414, 1624, 1840, 2106, 2384, 2717, 3070, 3492, 3936, 4464, 5026, 5684, 6388, 7210, 8088
Offset: 0

Views

Author

Olivier Gérard

Keywords

Comments

Case k=3, i=1 of Gordon Theorem.

Examples

			G.f. = 1 + x^2 + x^3 + 2*x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 5*x^8 + 6*x^9 + ...
G.f. = q^17 + q^101 + q^143 + 2*q^185 + 2*q^227 + 3*q^269 + 3*q^311 + ...

References

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

Links

Crossrefs

Cf. A035938, A035939.

Programs

  • Maple
    with (numtheory):
GordonsTheorem := proc(A, n) local L,M,m,i,s,d;
L := []; M := []; m := nops(A);
for i in [$1..n] do
    s := add(d*A[((d-1) mod m) + 1], d = divisors(i));
    L := [op(L), s];
    s := s + add(L[d]*M[i-d], d = [$1..i-1]);
    M := [op(M), s/i];
od; M end:
A035937_list := n -> GordonsTheorem([0, 1, 1, 1, 1, 0, 0], n):
A035937_list(40);  # Peter Luschny, Jan 22 2012
  • Mathematica
    f[max_][a_, b_] := Sum[a^(n*(n+1)/2)*b^(n*(n-1)/2), {n, -max, max}]; a[n_, max_] := a[n, max] = SeriesCoefficient[f[max][-x, -x^6]/f[max][-x, -x^2], {x, 0, n}]; a[n_] := (a[n, 2]; a[n, max = 3]; While[a[n, max] != a[n, max-1], max++]; a[n, max]); Table[a[n], {n, 0, 99}] (* Jean-François Alcover, Jan 13 2014 *)
a[ n_] := SeriesCoefficient[ 1 / Product[ (1 - x^(7 k - 2)) (1 - x^(7 k - 3)) (1 - x^(7 k - 4)) (1 - x^(7 k - 5)), {k, Ceiling[n/7]}], {x, 0, n}]; (* Michael Somos, Dec 30 2014 *)
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x^2, x^7] QPochhammer[ x^3, x^7] QPochhammer[ x^4, x^7] QPochhammer[ x^5, 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, 0, 1, 1, 1, 1, 0][k%7 + 1] * x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Dec 30 2014 */
  • Sage
    def GordonsTheorem(A, n) :
    L = []; M = [];
    m = len(A)
    for i in range(n) :
        s = sum(d*A[(d-1) % m] for d in divisors(i+1))
        L.append(s)
        s = s + sum(L[d-1]*M[i-d] for d in (1..i))
        M.append(s/(i+1))
    return M
def A035937_list(len) :  return GordonsTheorem([0, 1, 1, 1, 1, 0, 0], len)
A035937_list(40) # Peter Luschny, Jan 22 2012

Formula

Expansion of f(-x, -x^6) / f(-x, -x^2) in powers of x where f() is Ramanujan's general theta function.
Euler transform of period 7 sequence [ 0, 1, 1, 1, 1, 0, 0, ...]. - Michael Somos, Dec 30 2014
G.f.: 1 / (Product_{k>0} (1 - x^(7*k - 5)) * (1 - x^(7*k - 4)) * (1 - x^(7*k - 3)) * (1 - x^(7*k - 2))). - Michael Somos, Dec 30 2014 [corrected by Vaclav Kotesovec, Nov 12 2015]
G.f.: (Product_{k>1} (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 + x^(2*i+1)))). - Michael Somos, Dec 31 2014
a(n) ~ 2^(1/4) * sin(Pi/7) * exp(2*Pi*sqrt(2*n/21)) / (3^(1/4) * 7^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 12 2015

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

A100823 G.f.: Product_{k>0} (1+x^k)/((1-x^k)*(1+x^(3k))*(1+x^(5k))).

Original entry on oeis.org

1, 2, 4, 7, 12, 19, 30, 46, 69, 101, 146, 208, 293, 408, 563, 768, 1040, 1397, 1864, 2470, 3254, 4261, 5550, 7192, 9277, 11911, 15229, 19391, 24597, 31085, 39150, 49142, 61489, 76702, 95401, 118324, 146362, 180573, 222226, 272826, 334173, 408394, 498022
Offset: 0

Views

Author

Noureddine Chair, Jan 06 2005

Keywords

Comments

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

Examples

			G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 19*x^5 + 30*x^6 + 46*x^7 + ...
G.f. = q^-1 + 2*q^2 + 4*q^5 + 7*q^8 + 12*q^11 + 19*q^14 + 30*q^17 + 46*q^20 + ...

Links

Crossrefs

Cf. A035939, A098151, A102346, A122129.

Programs

  • Maple
    series(product((1+x^k)/((1-x^k)*(1+x^(3*k))*(1+x^(5*k))),k=1..100),x=0,100);
  • Mathematica
    CoefficientList[ Series[ Product[(1 + x^k)/((1 - x^k)*(1 + x^(3k))*(1 + x^(5k))), {k, 100}], {x, 0, 45}], x] (* Robert G. Wilson v, Jan 12 2005 *)
nmax = 50; CoefficientList[Series[Product[(1+x^(5*k-1))*(1+x^(5*k-2))*(1+x^(5*k-3))*(1+x^(5*k-4)) / ((1-x^(6*k))*(1-x^(3*k-1))*(1-x^(3*k-2))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 01 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3, x^6] QPochhammer[ x^5, x^10] / EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Mar 07 2016 *)
  • PARI
    q='q+O('q^33); E(k)=eta(q^k);
Vec( (E(2)*E(3)*E(5)) / (E(1)^2*E(6)*E(10)) ) \\ Joerg Arndt, Sep 01 2015

Formula

a(n) ~ exp(Pi*sqrt(37*n/5)/3) * sqrt(37) / (12*sqrt(5)*n). - Vaclav Kotesovec, Sep 01 2015
G.f.: (E(2)*E(3)*E(5)) / (E(1)^2*E(6)*E(10)) where E(k) = prod(n>=1, 1-q^k ). - Joerg Arndt, Sep 01 2015
Euler transform of period 30 sequence [ 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 0, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, ...]. - Michael Somos, Mar 07 2016
Expansion of chi(-x^3) * chi(-x^5) / phi(-x) in powers of x where phi(), chi() are Ramanujan theta functions. - Michael Somos, Mar 07 2016
a(n) - A035939(2*n + 1) = A122129(2*n + 1). - Michael Somos, Mar 07 2016

Extensions

More terms from Robert G. Wilson v, Jan 12 2005
Offset corrected by Vaclav Kotesovec, Sep 01 2015
a(14) = 563 <- 562 corrected by Vaclav Kotesovec, Sep 01 2015
Showing 1-3 of 3 results.

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