A036890 Number of partitions of 5n such that cn(1,5) = cn(4,5) < cn(0,5) <= cn(2,5) = cn(3,5).

0, 1, 4, 11, 27, 63, 142, 312, 665, 1382, 2795, 5524, 10674, 20228, 37634, 68886, 124179, 220779, 387458, 671883, 1152027, 1954614, 3283494, 5464437, 9013558, 14743397, 23923577, 38526121, 61593796, 97795238, 154251217, 241765892, 376643803, 583370176

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[3]=t[4] and t[5]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_List] := Module[{ll, mn, x}, ll = MapAt[#+1&, l, Mod[i, 5]+1]; mn = Min[ll]; If[mn == 0, ll, ll-mn]]; g[n_, i_, t_List] := g[n, i, t] = Which[n<0, 0, n == 0, If[t[[2]] == t[[5]] && t[[3]] == t[[4]] && t[[5]] < t[[1]] && t[[1]] <= t[[3]], 1, 0], True, Which[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, 2, 4, 4, 5}]]]], True, g[n, i, t] = g [n, i-1, t] + g[n-i, i, mkl[i, t]]]]; a[n_] := a[n] = g[5*n, 5*n, {0, 0, 0, 0, 0}]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, 32}] (* Jean-François Alcover, Dec 23 2015, after Alois P. Heinz *)

Formula

a(n) = A036892(n) + A036894(n).

a(n) = A036881(n) - A036884(n).

Extensions

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

a(33)-a(34) from Max Alekseyev, Dec 11 2011