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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. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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).

REFERENCES

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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

EXAMPLE

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 + ...

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 *)

PROG

(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

(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 */

CROSSREFS

Cf. A035937, A035939.

Sequence in context: A193748 A039852 A241068 * A024503 A061790 A107236

Adjacent sequences:  A035935 A035936 A035937 * A035939 A035940 A035941

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

STATUS

approved

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Last modified November 20 10:41 EST 2017. Contains 294963 sequences.