OFFSET
1,2
COMMENTS
Alternatively, number of partitions of 5n such that cn(0,5) <= cn(2,5) = cn(3,5) <= cn(1,5) = cn(4,5).
For a given partition, cn(i,n) means the number of its parts equal to i modulo n.
MAPLE
mkl:= proc(i, l) local ll, mn, x; ll:= applyop(x->x+1, irem(i, 5)+1, l); mn:= min(ll[]); `if`(mn=0, ll, map(x->x-mn, ll)) end:
g:= proc (n, i, t) if n<0 then 0 elif n=0 then `if`(t[1]<=t[2] and t[2]=t[5] and t[5]<=t[3] and t[3]=t[4], 1, 0) elif i=0 then 0 elif i=1 then g(0, 0, [t[1], t[2]+n, t[3], t[4], t[5]]) elif i=2 then `if`(t[3]>t[4], 0, g(n-2*(t[4]-t[3]), 1, [t[1], t[2], t[4], t[4], t[5]])) 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, 0]):
seq(a(n), n=1..15); # Alois P. Heinz, Jul 02 2009
MATHEMATICA
mkl[i_, l_] := Module[{ll, mn, x}, ll = MapAt[#+1&, l, Mod[i, 5]+1]; mn = Min[ll]; If[mn==0, ll, Map[#-mn&, ll]]]; g[n_, i_, t_List] := g[n, i, t] = Which[n<0, 0, n == 0 , If[t[[1]] <= t[[2]] && t[[2]] == t[[5]] && t[[5]] <= t[[3]] && t[[3]] == t[[4]], 1, 0], i==0, 0, i==1, g[0, 0, {t[[1]], t[[2]]+n, t[[3]], t[[4]], t[[5]]}] , i==2, If[t[[3]]>t[[4]], 0, g[n-2*(t[[4]]-t[[3]]), 1, {t[[1]], t[[2]], t[[4]], t[[4]], t[[5]]}]], True, g[n, i-1, t] + g[n-i, i, mkl[i, t]]]; a[n_] := g[5*n, 5*n, {0, 0, 0, 0, 0}]; Table[a[n], {n, 1, 15}] (* Jean-François Alcover, Jul 29 2015, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
a(10)-a(31) from Alois P. Heinz, Jul 02 2009
Edited by Max Alekseyev, Dec 11 2011
a(32)-a(33) from Alois P. Heinz, Mar 12 2016
STATUS
approved