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A237833 Number of partitions of n such that (greatest part) - (least part) > number of parts. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,7

LINKS

R. J. Mathar, Table of n, a(n) for n = 1..96

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

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 *)

CROSSREFS

Cf. A237830, A237831, A237832, A237834.

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

Sequence in context: A147789 A047625 A147871 * A275633 A004397 A324368

Adjacent sequences:  A237830 A237831 A237832 * A237834 A237835 A237836

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Feb 16 2014

STATUS

approved

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Last modified November 25 19:24 EST 2020. Contains 338625 sequences. (Running on oeis4.)