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
Keywords
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
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
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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
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
Comments