|
|
A035985
|
|
Number of partitions of n into parts not a multiple of 7. Also number of partitions with at most 6 parts of size 1 and differences between parts at distance 9 are greater than 1.
|
|
16
|
|
|
1, 1, 2, 3, 5, 7, 11, 14, 21, 28, 39, 51, 70, 90, 119, 153, 199, 252, 324, 406, 515, 642, 804, 994, 1236, 1517, 1869, 2282, 2791, 3387, 4118, 4970, 6006, 7217, 8673, 10374, 12411, 14780, 17601, 20883, 24766, 29274, 34588, 40741, 47964, 56319, 66080, 77350
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,3
|
|
COMMENTS
|
Case k=10, i=7 of Gordon Theorem.
|
|
REFERENCES
|
G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.
|
|
LINKS
|
|
|
FORMULA
|
Euler transform of period 7 sequence [1, 1, 1, 1, 1, 1, 0, ...]. - Michael Somos, Jan 17 2006
Given g.f. A(x), then B(x)=x*A(x^4) satisfies 0=f(B(x), B(x^3)) where f(u, v)=(u^4+v^4)-u*v*(1+3*u*v+7*(u*v)^2).
G.f.: Product_{k>0} (1-x^(7k))/(1-x^k).
Given g.f. A(x) then B(x)=x*A(x)^4 satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u,v,w)= (u^2+u*w+w^2) -v -8*v*(u+v+w) -49*v^2*(u+w). - Michael Somos, May 28 2006
G.f. is product k>0 P7(x^k) where P7 is 7th cyclotomic polynomial.
Expansion of q^(-1/4)eta(q^7)/eta(q) in powers of q. - Michael Somos, Jan 17 2006
a(n) ~ 2*Pi * BesselI(1, sqrt((4*n + 1)/7) * Pi) / (7*sqrt(4*n + 1)) ~ exp(2*Pi*sqrt(n/7)) / (2 * 7^(3/4) * n^(3/4)) * (1 + (Pi/(4*sqrt(7)) - 3*sqrt(7)/(16*Pi)) / sqrt(n) + (Pi^2/224 - 105/(512*Pi^2) - 15/64) / n). - Vaclav Kotesovec, Aug 31 2015, extended Jan 14 2017
|
|
EXAMPLE
|
B(x) = x +x^5 +2*x^9 +3*x^13 +5*x^17 +7*x^21 +11*x^25 +14*x^29 +...
|
|
MATHEMATICA
|
nmax = 50; CoefficientList[Series[Product[(1 - x^(7*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
QP = QPochhammer; s = QP[q^7]/QP[q] + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)
Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 7], 0, 2] ], {n, 0, 47}] (* Robert Price, Jul 28 2020 *)
|
|
PROG
|
(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^7+A)/eta(x+A), n))} /* Michael Somos, Jan 17 2006 */
(PARI) Vec(prod(k=1, 50, (1 - x^(7*k))/(1 - x^k)) + O(x^51)) \\ Indranil Ghosh, Mar 25 2017
|
|
CROSSREFS
|
Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|