A387739 Number of partitions of n such that the largest part is >= n/2 and each part is congruent to 1 or 4 (mod 5).

1, 1, 0, 1, 1, 2, 2, 3, 2, 3, 4, 5, 3, 4, 6, 7, 9, 11, 8, 11, 13, 16, 11, 14, 18, 22, 26, 32, 24, 30, 36, 43, 32, 38, 47, 55, 66, 77, 61, 73, 86, 101, 77, 92, 108, 127, 147, 172, 138, 162, 188, 217, 172, 199, 232, 267, 308, 353, 290, 334, 384, 439, 353, 407

Offset

1,6

Links

Sean A. Irvine, Table of n, a(n) for n = 1..1000

Formula

a(n) = Sum_{k=ceiling(n/2)..n,k={1,4}(mod 5)} B(n-k,k) where B(0,k)=1, B(n,k)=0 if n<0 or k<=0, B(n,k) = B(n,k-1) + B(n-k,k) if k = 1 or 4 (mod 5), otherwise B(n,k) = B(n,k-1).

Example

a(6)=2 because of the partitions 6 and 4+1+1.

Mathematica

a[n_]:=Module[{p, w, T, r, w2}, p=Select[Range[n], Mod[#, 5]==1||Mod[#, 5]==4&];
w=Table[0, n+1]; w[[1]]=1;
Do[If[s>=t, w[[s+1]]+=w[[s-t+1]]], {t, p}, {s, 0, n}];
T=Floor[(n-1)/2];
r=Select[p, #<=T&];
w2=Table[0, n+1]; w2[[1]]=1;
Do[If[s>=t, w2[[s+1]]+=w2[[s-t+1]]], {t, r}, {s, 0, n}];
w[[n+1]]-w2[[n+1]]]
Table[a[n], {n, 1, 100}] (* Vincenzo Librandi, Dec 01 2025 *)

Prog

(Magma) function a(n)
    p := [k : k in [1..n] | (k mod 5) in {1, 4}];
    w := [0 : i in [0..n]]; w[1] := 1;
    for t in p do
        for s in [t..n] do
            w[s+1] +:= w[s-t+1];
        end for;
    end for;
    T := (n-1) div 2;
    r := [k : k in p | k le T];
    w2 := [0 : i in [0..n]]; w2[1] := 1;
    for t in r do
        for s in [t..n] do
            w2[s+1] +:= w2[s-t+1];
        end for;
    end for;
    return w[n+1] - w2[n+1];
end function;
seq := [ a(n) : n in [1..100] ];
print seq; // Vincenzo Librandi, Dec 01 2025

Crossrefs

Cf. A003114, A387735, A387736, A387737, A387738, A387740.

Sequence in context: A350511 A324344 A059711 * A318287 A048621 A308048

Adjacent sequences: A387736 A387737 A387738 * A387740 A387741 A387742

Keyword

nonn

Author

Sean A. Irvine, Sep 07 2025

Status

approved