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

Author

Olivier Gérard

Keywords

• nonn
• easy

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

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

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