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A237832
Number of partitions of n such that (greatest part) - (least part) = number of parts.
17
0, 0, 0, 1, 0, 2, 1, 3, 3, 5, 4, 10, 8, 13, 15, 22, 22, 34, 36, 51, 58, 75, 85, 116, 130, 165, 194, 244, 281, 355, 409, 505, 591, 718, 839, 1022, 1186, 1425, 1668, 1994, 2319, 2765, 3213, 3805, 4429, 5214, 6052, 7119, 8243, 9645, 11169, 13026, 15046, 17511
OFFSET
1,6
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
A237830(n) + a(n) + A237833(n) = A000041(n). - R. J. Mathar, Nov 24 2017
G.f.: (1/Product_{k>=1} (1-x^k)) * (1/(1-x)) * Sum_{k>=1} (-1)^(k-1) * ( k * (1-x) * x^(k*(3*k-1)/2) * (1+x^k) - x^(3*k*(k-1)/2+1) * (1-x^(2*k)) ) - Seiichi Manyama, May 20 2023
EXAMPLE
a(6) = 2 counts these partitions: 4+2, 4+1+1.
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=60, x='x+O('x^N)); concat([0, 0, 0], Vec(1/prod(k=1, N, 1-x^k)*1/(1-x)*sum(k=1, N, (-1)^(k-1)*(k*(1-x)*x^(k*(3*k-1)/2)*(1+x^k)-x^(3*k*(k-1)/2+1)*(1-x^(2*k)))))) \\ Seiichi Manyama, May 20 2023
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Clark Kimberling, Feb 16 2014
STATUS
approved