A209354 - OEIS

%I #8 Oct 13 2023 06:48:52

%S 1,0,1,1,0,1,1,1,0,1,2,1,1,0,1,1,3,1,1,0,1,3,2,3,1,1,0,1,1,5,3,3,1,1,

%T 0,1,3,4,6,3,3,1,1,0,1,2,6,6,7,3,3,1,1,0,1,3,6,10,7,7,3,3,1,1,0,1,1,9,

%U 10,12,8,7,3,3,1,1,0,1,5,6,15,14,13,8,7,3,3,1,1,0,1,1,11,15,20

%N Triangular array:  T(n,k) = number of partitions of n for which (maximal term)-(minimal term)=k, if 0<=k<n, and T(n,n)=1.

%C Row sums: A000041 (number of partitions of n).

%e First ten rows:

%e   1

%e   0 1

%e   1 0 1

%e   1 1 0 1

%e   2 1 1 0 1

%e   1 3 1 1 0 1

%e   3 2 3 1 1 0 1

%e   1 5 3 3 1 1 0 1

%e   3 4 6 3 3 1 1 0 1

%e   2 6 6 7 3 3 1 1 0 1

%e Row 5 (counting the top row as row 0):

%e   T(5,0)=1 counts [1,1,1,1,1]

%e   T(5,1)=3 counts [2,1,1,1], [2,2,1], [3,2]

%e   T(5,2)=1 counts [3,1,1]

%e   T(5,3)=1 counts [4,1]

%e   T(5,4)=0

%e   T(5,5)=1 counts [5]

%t f[n_] := IntegerPartitions[n];

%t p[n_, k_] := f[n][[k]];

%t r[n_] := Table[p[n, k], {k, 1, Length[f[n]]}]

%t g[n_, k_] := Max[p[n, k]] - Min[p[n, k]]; g[n_, 1] := n;

%t t[n_] := Table[g[n, k], {k, 1, Length[f[n]]}]

%t c[0, 0] = 1; c[n_, k_] := Count[t[n], k]

%t u = Table[c[n, k], {n, 0, 15}, {k, 0, n}];

%t TableForm[u] (* as a triangle *)

%t Flatten[u]   (* as a sequence *)

%Y Cf. A000041.

%K nonn,tabl

%O 1,11

%A _Clark Kimberling_, Mar 06 2012