A035937 - OEIS

%I #41 Jan 12 2019 20:43:42

%S 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,

%T 161,186,221,254,301,344,405,464,541,619,720,820,949,1081,1245,1414,

%U 1624,1840,2106,2384,2717,3070,3492,3936,4464,5026,5684,6388,7210,8088

%N 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.

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

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

%H Jean-François Alcover and Vaclav Kotesovec, <a href="/A035937/b035937.txt">Table of n, a(n) for n = 0..1000</a> (first 99 terms from Jean-François Alcover)

%F Expansion of f(-x, -x^6) / f(-x, -x^2) in powers of x where f() is Ramanujan's general theta function.

%F Euler transform of period 7 sequence [ 0, 1, 1, 1, 1, 0, 0, ...]. - _Michael Somos_, Dec 30 2014

%F 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]

%F 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

%F 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

%e 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 + ...

%e G.f. = q^17 + q^101 + q^143 + 2*q^185 + 2*q^227 + 3*q^269 + 3*q^311 + ...

%p with (numtheory):

%p GordonsTheorem := proc(A, n) local L,M,m,i,s,d;

%p L := []; M := []; m := nops(A);

%p for i in [$1..n] do

%p     s := add(d*A[((d-1) mod m) + 1], d = divisors(i));

%p     L := [op(L), s];

%p     s := s + add(L[d]*M[i-d], d = [$1..i-1]);

%p     M := [op(M), s/i];

%p od; M end:

%p A035937_list := n -> GordonsTheorem([0, 1, 1, 1, 1, 0, 0], n):

%p A035937_list(40);  # _Peter Luschny_, Jan 22 2012

%t 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 *)

%t 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 *)

%t 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 *)

%o (Sage)

%o def GordonsTheorem(A, n) :

%o     L = []; M = [];

%o     m = len(A)

%o     for i in range(n) :

%o         s = sum(d*A[(d-1) % m] for d in divisors(i+1))

%o         L.append(s)

%o         s = s + sum(L[d-1]*M[i-d] for d in (1..i))

%o         M.append(s/(i+1))

%o     return M

%o def A035937_list(len) :  return GordonsTheorem([0, 1, 1, 1, 1, 0, 0], len)

%o A035937_list(40) # Peter Luschny, Jan 22 2012

%o (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 */

%Y Cf. A035938, A035939.

%K nonn,easy

%O 0,5

%A _Olivier Gérard_