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

0,5

Comments

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

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)

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

Example

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 + ...

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 *)

Prog

(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
(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 */

Crossrefs

Cf. A035938, A035939.

Sequence in context: A035371 A035577 A002723 * A240201 A274158 A020999

Adjacent sequences: A035934 A035935 A035936 * A035938 A035939 A035940

Keyword

nonn,easy

Author

Olivier Gérard

Status

approved