A202192 Number of partitions of 5n with equal number of parts congruent to each of 1, 2, 3 and 4 modulo 5.

1, 1, 3, 8, 22, 53, 124, 269, 568, 1152, 2284, 4410, 8363, 15542, 28438, 51201, 90930, 159300, 275740, 471706, 798388, 1337478, 2219395, 3649432, 5950078, 9622364, 15442269, 24600952, 38919910, 61164114, 95513618, 148247892, 228761668, 351032568, 535772894

Offset: 0

Author

Max Alekseyev, Dec 14 2011

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100
• Index and properties of sequences related to partitions of 5n

Crossrefs

Cf. A046776.

Programs

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}] + PartitionsP[n];
Table[a[n], {n, 0, 34}] (* Jean-François Alcover, May 25 2019, after Alois P. Heinz in A046787 *)

Formula

a(n) = A046776(n) + A202086(n) + A202088(n).

a(n) = A046787(n) + A000041(n).

Extensions

a(33)-a(34) from Alois P. Heinz, May 24 2019