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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. 5
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified January 16 10:47 EST 2021. Contains 340204 sequences. (Running on oeis4.)