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

0,4

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

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

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 *)

Prog

(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

Crossrefs

Other similar sequences include:

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

Mod 5: A046783, A046784, A046785, A046786.

Cf. A046765, A046776, A202192.

Sequence in context: A147043 A146264 A146216 * A003295 A228857 A253427

Adjacent sequences: A046784 A046785 A046786 * A046788 A046789 A046790

Keyword

nonn

Author

David W. Wilson

Extensions

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

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

Status

approved