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

%I A237833 #30 Dec 18 2023 10:09:03
%S A237833 0,0,0,0,1,1,3,4,7,10,16,20,31,41,56,74,101,129,172,219,284,362,463,
%T A237833 579,735,918,1147,1422,1767,2172,2680,3279,4013,4888,5947,7200,8721,
%U A237833 10515,12663,15202,18235,21798,26039,31015,36898,43802,51930,61426,72590
%N A237833 Number of partitions of n such that (greatest part) - (least part) > number of parts.
%H A237833 Seiichi Manyama, <a href="/A237833/b237833.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..96 from R. J. Mathar)
%H A237833 George E. Andrews, <a href="https://georgeandrews1.github.io/pdf/315.pdf">4-Shadows in q-Series and the Kimberling Index</a>, Preprint, May 15, 2016.
%F A237833 A237831(n) + a(n) = A000041(n). - _R. J. Mathar_, Nov 24 2017
%F A237833 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
%e A237833 a(8) = 4 counts these partitions:  7+1, 6+2, 6+1+1, 5+2+1.
%p A237833 isA237833 := proc(p)
%p A237833     if abs(p[1]-p[-1]) > nops(p) then
%p A237833         return 1;
%p A237833     else
%p A237833         return 0;
%p A237833     end if;
%p A237833 end proc:
%p A237833 A237833 := proc(n)
%p A237833     local a,p;
%p A237833     a := 0 ;
%p A237833     p := combinat[firstpart](n) ;
%p A237833     while true do
%p A237833         a := a+isA237833(p) ;
%p A237833         if nops(p) = 1 then
%p A237833             break;
%p A237833         end if;
%p A237833         p := nextpart(p) ;
%p A237833     end do:
%p A237833     return a;
%p A237833 end proc:
%p A237833 seq(A237833(n),n=1..20) ; # _R. J. Mathar_, Nov 17 2017
%t A237833 z = 60; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := t[p] = Length[p];
%t A237833 Table[Count[q[n], p_ /; Max[p] - Min[p] < t[p]], {n, z}]  (* A237830 *)
%t A237833 Table[Count[q[n], p_ /; Max[p] - Min[p] <= t[p]], {n, z}] (* A237831 *)
%t A237833 Table[Count[q[n], p_ /; Max[p] - Min[p] == t[p]], {n, z}] (* A237832 *)
%t A237833 Table[Count[q[n], p_ /; Max[p] - Min[p] > t[p]], {n, z}]  (* A237833 *)
%t A237833 Table[Count[q[n], p_ /; Max[p] - Min[p] >= t[p]], {n, z}] (* A237834 *)
%o A237833 (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
%Y A237833 Cf. A237830, A237831, A237832, A237834.
%Y A237833 Different from, but has the same beginning as, A275633.
%K A237833 nonn,easy
%O A237833 1,7
%A A237833 _Clark Kimberling_, Feb 16 2014