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Number of partitions satisfying cn(1,5) = cn(4,5) = 0.
5

%I #31 Aug 10 2020 08:44:05

%S 0,1,1,1,2,2,3,4,4,7,6,10,11,13,18,19,25,30,33,45,47,61,70,81,100,111,

%T 135,157,177,218,238,288,328,374,443,495,579,663,747,878,973,1134,

%U 1281,1448,1670,1863,2135,2414,2705,3103

%N Number of partitions satisfying cn(1,5) = cn(4,5) = 0.

%C For a given partition cn(i,n) means the number of its parts equal to i modulo n.

%C Short: (1=4 := 0).

%C a(n) is the number of partitions with parts congruent to 0, 2 or 3 mod 5. - _George Beck_, Aug 08 2020

%H Jinyuan Wang, <a href="/A036822/b036822.txt">Table of n, a(n) for n = 1..1000</a>

%F Convolution inverse of A113428. - _George Beck_, May 21 2016

%F G.f.: Product_{k>=1} 1/((1 - x^(5*k)) * (1 - x^(5*k - 2)) * (1 - x^(5*k - 3))). - _Vaclav Kotesovec_, Jul 05 2016

%F a(n) ~ exp(Pi*sqrt(2*n/5)) / (2*sqrt(2*(5+sqrt(5)))*n). - _Vaclav Kotesovec_, Jul 05 2016

%p c := proc(L,i,n)

%p local a,p;

%p a := 0 ;

%p for p in L do

%p if modp(p,n) = i then

%p a := a+1 ;

%p end if;

%p end do:

%p a ;

%p end proc:

%p A036822 := proc(n)

%p local a ,p;

%p a := 0 ;

%p for p in combinat[partition](n) do

%p if c(p,1,5) = 0 then

%p if c(p,4,5) = 0 then

%p a := a+1 ;

%p end if;

%p end if;

%p end do:

%p a ;

%p end proc: # _R. J. Mathar_, Oct 19 2014

%t nmax = 50; Rest[CoefficientList[Series[Product[1/((1 - x^(5*k)) * (1 - x^(5*k-2)) * (1 - x^(5*k-3))), {k, 1, nmax}], {x, 0, nmax}], x]] (* _Vaclav Kotesovec_, Jul 05 2016 *)

%Y Cf. A036820.

%K nonn

%O 1,5

%A _Olivier GĂ©rard_