A036822 Number of partitions satisfying cn(1,5) = cn(4,5) = 0.

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, 135, 157, 177, 218, 238, 288, 328, 374, 443, 495, 579, 663, 747, 878, 973, 1134, 1281, 1448, 1670, 1863, 2135, 2414, 2705, 3103

Offset: 1

Author

Olivier Gérard

Keywords

• nonn

Comments

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

Short: (1=4 := 0).

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

Links

• Jinyuan Wang, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A036820.

Programs

Maple

c := proc(L,i,n)
    local a,p;
    a := 0 ;
    for p in L do
        if modp(p,n) = i then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
A036822 := proc(n)
    local a ,p;
    a := 0 ;
    for p in combinat[partition](n) do
        if c(p,1,5) = 0 then
            if c(p,4,5) = 0 then
                a := a+1 ;
            end if;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Oct 19 2014

Mathematica

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 *)

Formula

Convolution inverse of A113428. - George Beck, May 21 2016

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

a(n) ~ exp(Pi*sqrt(2*n/5)) / (2*sqrt(2*(5+sqrt(5)))*n). - Vaclav Kotesovec, Jul 05 2016