A237833 Number of partitions of n such that (greatest part) - (least part) > number of parts.

0, 0, 0, 0, 1, 1, 3, 4, 7, 10, 16, 20, 31, 41, 56, 74, 101, 129, 172, 219, 284, 362, 463, 579, 735, 918, 1147, 1422, 1767, 2172, 2680, 3279, 4013, 4888, 5947, 7200, 8721, 10515, 12663, 15202, 18235, 21798, 26039, 31015, 36898, 43802, 51930, 61426, 72590

Offset: 1

Author

Clark Kimberling, Feb 16 2014

Keywords

• nonn
• easy

Examples

			a(8) = 4 counts these partitions:  7+1, 6+2, 6+1+1, 5+2+1.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..96 from R. J. Mathar)
• George E. Andrews, 4-Shadows in q-Series and the Kimberling Index, Preprint, May 15, 2016.

Crossrefs

Cf. A237830, A237831, A237832, A237834.

Different from, but has the same beginning as, A275633.

Programs

Maple

isA237833 := proc(p)
    if abs(p[1]-p[-1]) > nops(p) then
        return 1;
    else
        return 0;
    end if;
end proc:
A237833 := proc(n)
    local a,p;
    a := 0 ;
    p := combinat[firstpart](n) ;
    while true do
        a := a+isA237833(p) ;
        if nops(p) = 1 then
            break;
        end if;
        p := nextpart(p) ;
    end do:
    return a;
end proc:
seq(A237833(n),n=1..20) ; # R. J. Mathar, Nov 17 2017

Mathematica

z = 60; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := t[p] = Length[p];
Table[Count[q[n], p_ /; Max[p] - Min[p] < t[p]], {n, z}]  (* A237830 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] <= t[p]], {n, z}] (* A237831 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] == t[p]], {n, z}] (* A237832 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] > t[p]], {n, z}]  (* A237833 *)
Table[Count[q[n], p_ /; Max[p] - Min[p] >= t[p]], {n, z}] (* A237834 *)

PARI

my(N=50, x='x+O('x^N)); concat([0, 0, 0, 0], Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^k*(k-1)*(x^(k*(3*k-1)/2)+x^(k*(3*k+1)/2))))) \\ Seiichi Manyama, May 20 2023

Formula

A237831(n) + a(n) = A000041(n). - R. J. Mathar, Nov 24 2017

G.f.: (1/Product_{k>=1} (1-x^k)) * Sum_{k>=1} (-1)^k * (k-1) * ( x^(k*(3*k-1)/2) + x^(k*(3*k+1)/2) ). (See Andrews' preprint.) - Seiichi Manyama, May 20 2023