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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

%I

%S 1,1,2,3,5,7,11,14,21,28,39,51,70,90,119,153,199,252,324,406,515,642,

%T 804,994,1236,1517,1869,2282,2791,3387,4118,4970,6006,7217,8673,10374,

%U 12411,14780,17601,20883,24766,29274,34588,40741,47964,56319,66080,77350

%N 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.

%C Case k=10, i=7 of Gordon Theorem.

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

%H Seiichi Manyama, <a href="/A035985/b035985.txt">Table of n, a(n) for n = 0..10000</a>

%H GDZ, <a href="http://www.digizeitschriften.de/dms/toc/?PID=PPN243919689">Digitized volumes of Crelle</a>

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 15.

%H G. N. Watson, <a href="http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002174499">Ramanujans Vermutung über Zerfällungsanzahlen</a>. J. Reine Angew. Math. (Crelle), 179 (1938), 97-128. This sequence arises as the coefficients of Y = C/B on p. 118.

%F Euler transform of period 7 sequence [1, 1, 1, 1, 1, 1, 0, ...]. - _Michael Somos_, Jan 17 2006

%F 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).

%F G.f.: Product_{k>0} (1-x^(7k))/(1-x^k).

%F 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

%F G.f. is product k>0 P7(x^k) where P7 is 7th cyclotomic polynomial.

%F Expansion of q^(-1/4)eta(q^7)/eta(q) in powers of q. - _Michael Somos_, Jan 17 2006

%F 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

%F a(n) = (1/n)*Sum_{k=1..n} A113957(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 25 2017

%e B(x) = x +x^5 +2*x^9 +3*x^13 +5*x^17 +7*x^21 +11*x^25 +14*x^29 +...

%t nmax = 50; CoefficientList[Series[Product[(1 - x^(7*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 31 2015 *)

%t QP = QPochhammer; s = QP[q^7]/QP[q] + O[q]^50; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 30 2015, adapted from PARI *)

%t Table[Count[IntegerPartitions@n, x_ /; ! MemberQ [Mod[x, 7], 0, 2] ], {n, 0, 47}] (* _Robert Price_, Jul 28 2020 *)

%o (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 */

%o (PARI) Vec(prod(k=1, 50, (1 - x^(7*k))/(1 - x^k)) + O(x^51)) \\ _Indranil Ghosh_, Mar 25 2017

%o (PARI) A035985_upto(N,q='x+O('x^N))=Vec(eta(q^7)/eta(q)) \\ _M. F. Hasler_, Dec 09 2019

%Y Number of r-regular partitions for r = 2 through 12: A000009, A000726, A001935, A035959, A219601, A035985, A261775, A104502, A261776, A328545, A328546.

%Y Cf. A320609.

%K nonn,easy

%O 0,3

%A _Olivier Gérard_

%E Definition simplified by _N. J. A. Sloane_, Oct 20 2019

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Last modified August 5 22:41 EDT 2021. Contains 346488 sequences. (Running on oeis4.)