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