A240872 Number of partitions p of n into distinct parts such that max(p) = 4 + min(p).

0, 0, 0, 0, 0, 0, 1, 0, 2, 1, 2, 2, 3, 2, 2, 3, 3, 2, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 2, 2, 4, 2, 2, 2, 3, 3, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 3, 2, 2, 2, 4

Offset: 0

Author

Clark Kimberling, Apr 15 2014

Keywords

• nonn
• easy

Examples

			a(12) counts these 3 partitions:  84, 642, 5421.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (-1,-1,-1,0,1,1,1,1).

Crossrefs

Cf. A171182, A240871, A240873.

Programs

Mathematica

    z = 40; f[n_] := f[n] = Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &];
    Table[Count[f[n], p_ /; Max[p] == 2 + Min[p]], {n, 0, z}] (* A171182 *)
    Table[Count[f[n], p_ /; Max[p] == 3 + Min[p]], {n, 0, z}] (* A240871 *)
    Table[Count[f[n], p_ /; Max[p] == 4 + Min[p]], {n, 0, z}] (* A240872 *)
    Table[Count[f[n], p_ /; Max[p] == 5 + Min[p]], {n, 0, z}] (* A240873 *)

Formula

G.f.: -x^6*(x^12+2*x^11+3*x^10+5*x^9+5*x^8+6*x^7+7*x^6+6*x^5+5*x^4+4*x^3+3*x^2+x+1) / ((x-1)*(x+1)*(x^2+1)*(x^4+x^3+x^2+x+1)). - Alois P. Heinz, Jun 18 2025