A046787 Number of partitions of 5n with equal nonzero number of parts congruent to each of 1, 2, 3 and 4 modulo 5.

0, 0, 1, 5, 17, 46, 113, 254, 546, 1122, 2242, 4354, 8286, 15441, 28303, 51025, 90699, 159003, 275355, 471216, 797761, 1336686, 2218393, 3648177, 5948503, 9620406, 15439833, 24597942, 38916192, 61159549, 95508014, 148241050, 228753319, 351022425, 535760584

Offset: 0

Author

David W. Wilson

Keywords

• nonn

Comments

Number of partitions of m with equal numbers of parts congruent to each of 1, 2, 3 and 4 (mod 5) is 0 unless m == 0 mod 5.

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000 (terms n=0..100 from Alois P. Heinz)
• Index and properties of sequences related to partitions of 5n

Crossrefs

Other similar sequences include:

Mod 4: A046778, A046779, A046780, A046781, A046782.

Mod 5: A046783, A046784, A046785, A046786.

Cf. A046765, A046776, A202192.

Programs

Maple

mkl:= proc(i,l) local ll, mn, x; ll:= `if`(irem(i, 5)=0, l, applyop(x->x+1, irem(i,5), l)); mn:= min(l[])-1; `if`(mn<=0, ll, map(x->x-mn, ll)) end:
g:= proc(n,i,t) local m, mx; if n<0 then 0 elif n=0 then `if`(t[1]>0 and t[1]=t[2] and t[2]=t[3] and t[3]=t[4], 1, 0) elif i=0 then 0 elif i<5 then mx:= max(t[]); m:= n-10*mx +t[1] +t[2]*2 +t[3]*3 +t[4]*4; `if`(m>=0 and irem(m, 10)=0, 1, 0) else g(n,i,t):= g(n, i-1, t) + g(n-i, i, mkl(i, t)) fi end:
a:= n-> g(5*n, 5*n, [0,0,0,0]):
seq(a(n), n=0..20);  # Alois P. Heinz, Jul 04 2009

Mathematica

mkl[i_, l_] := Module[{ll, mn, x}, ll = If[Mod[i, 5] == 0, l, MapAt[#+1&, l, Mod[i, 5]]]; mn = Min[l]-1; If[mn <= 0, ll, Map[#-mn&, ll]]];
g[n_, i_, t_] := g[n, i, t] = Module[{m, mx}, If[n<0, 0, If[n==0, If[ t[[1]]>0 && Equal @@ t[[1;;4]], 1, 0], If[i==0, 0, If[i<5, mx = Max[t]; m = n - 10 mx + t[[1]] + 2 t[[2]] + 3 t[[3]] + 4 t[[4]]; If[m >= 0 && Mod[m, 10]==0, 1, 0], g[n, i-1, t] + g[n-i, i, mkl[i, t]]]]]]];
a[n_] := g[5n, 5n, {0, 0, 0, 0}];
Table[a[n], {n, 0, 34}] (* Jean-François Alcover, May 25 2019, after Alois P. Heinz *)

PARI

seq(n)={Vec(sum(k=1, n\2, x^(2*k)/prod(j=1, k, 1 - x^j + O(x*x^(n-2*k)))^4)/prod(j=1, n, 1 - x^j + O(x*x^n)), -(n+1))} \\ Andrew Howroyd, Sep 16 2019

Formula

a(n) = A046776(n) + A202086(n) + A202088(n) - A000041(n) = A202192(n) - A000041(n). - Max Alekseyev

G.f.: (Sum_{k>0} x^(2*k)/(Product_{j=1..k} 1 - x^j)^4)/(Product_{j>0} 1 - x^j). - Andrew Howroyd, Sep 16 2019

Extensions

a(17)-a(32) from Alois P. Heinz, Jul 04 2009

a(33)-a(34) from Alois P. Heinz, Aug 13 2013