A036881 Number of partitions of 5n such that cn(1,5) = cn(4,5) <= cn(0,5) <= cn(2,5) = cn(3,5).

1, 4, 11, 29, 69, 160, 349, 743, 1526, 3067, 6011, 11566, 21813, 40476, 73879, 132927, 235842, 413211, 715261, 1224476, 2074156, 3479110, 5781362, 9523182, 15556055, 25210722, 40550228, 64757269, 102708208, 161838160, 253415308, 394437255

Offset: 1

Author

Olivier Gérard

Keywords

• nonn

Comments

Alternatively, number of partitions of 5n such that cn(2,5) = cn(3,5) <= cn(0,5) <= cn(1,5) = cn(4,5).

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

Links

• Index and properties of sequences related to partitions of 5n

Programs

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[2]=t[5] and t[5]<=t[1] and t[1]<=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[[2]] == t[[5]] && t[[5]] <= t[[1]] && t[[1]] <= 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 *)

Formula

a(n) = A036889(n) + A036885(n)

a(n) = A036884(n) + A036890(n)

Extensions

a(10)-a(32) from Alois P. Heinz, Jul 02 2009

Edited by Max Alekseyev, Dec 11 2011