A035938 -id:A035938 - cp's OEIS frontend

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.

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

Olivier Gérard

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

			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 + ...

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

Jean-François Alcover and Vaclav Kotesovec, Table of n, a(n) for n = 0..1000 (first 99 terms from Jean-François Alcover)

Cf. A035938, A035939.

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

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 *)

{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 */

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

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

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

Olivier Gérard

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

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

			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 + ...

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A035937, A035938.

# See A035937 for GordonsTheorem
A035939_list := n -> GordonsTheorem([1, 1, 0, 0, 1, 1, 0], n):
A035939_list(40); # Peter Luschny, Jan 22 2012

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 *)

{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 */

# 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

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

Missing a(0)=1 prepended by Michael Somos, Dec 29 2014

Name simplified by George Beck, Aug 27 2023

cp's OEIS Frontend

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.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Sage

Formula