A387740 Number of partitions of n such that the largest part is >= n/2 and each part is congruent to 2 or 3 (mod 5).

0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 4, 5, 4, 5, 3, 4, 5, 7, 8, 10, 12, 15, 13, 14, 10, 12, 15, 19, 22, 27, 32, 38, 31, 37, 27, 32, 38, 46, 53, 63, 74, 86, 72, 82, 63, 74, 86, 101, 117, 136, 157, 181, 151, 175, 136, 157, 181, 209, 239, 275, 314, 358, 304, 344

Offset

1,10

Links

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

Formula

a(n) = Sum_{k=ceiling(n/2)..n,k={2,3}(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 = 2 or 3 (mod 5), otherwise B(n,k) = B(n,k-1).

Example

a(10)=2 because of the partitions 8+2 and 7+3.

Mathematica

a[n_]:=Module[{p, w, T, r, w2}, p=Select[Range[n], Mod[#, 5]==2||Mod[#, 5]==3&];
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, Nov 30 2025 *)

Prog

(Magma) function a(n)
    p := [k : k in [1..n] | (k mod 5) in {2, 3}];
    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, Nov 30 2025

Crossrefs

Cf. A003106, A387735, A387736, A387737, A387738, A387739.

Sequence in context: A318560 A030385 A253553 * A031250 A031233 A030584

Adjacent sequences: A387737 A387738 A387739 * A387741 A387742 A387743

Keyword

nonn

Author

Sean A. Irvine, Sep 07 2025

Status

approved