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A035939 Number of partitions of n into parts not of the form 7k, 7k+3 or 7k-3. Also number of partitions with at most 2 parts of size 1 and differences between parts at distance 2 are greater than 1. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Case k=3, i=3 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

G. C. Greubel, 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^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

EXAMPLE

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 + ...

MAPLE

# See A035937 for GordonsTheorem

A035939_list := n -> GordonsTheorem([1, 1, 0, 0, 1, 1, 0], n):

A035939_list(40); # Peter Luschny, Jan 22 2012

MATHEMATICA

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 *)

PROG

(Sage) # 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

(PARI) {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 */

CROSSREFS

Cf. A035937, A035938.

Sequence in context: A238588 A102464 A082538 * A116665 A122135 A339572

Adjacent sequences:  A035936 A035937 A035938 * A035940 A035941 A035942

KEYWORD

nonn,easy

AUTHOR

Olivier Gérard

EXTENSIONS

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

STATUS

approved

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Last modified January 19 16:16 EST 2021. Contains 340270 sequences. (Running on oeis4.)