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

1,7

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.

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A237830, A237831, A237832, A237834.

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

Sequence in context: A047625 A147871 A368896 * A275633 A004397 A324368

Adjacent sequences: A237830 A237831 A237832 * A237834 A237835 A237836

Keyword

nonn,easy

Author

Clark Kimberling, Feb 16 2014

Status

approved