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A238406
Number T(n,k) of partitions of n into k parts such that every i-th smallest part (counted with multiplicity) is different from i; triangle T(n,k), n>=0, 0<=k<=floor((sqrt(9+8*n)-3)/2) read by rows.
10
1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 2, 0, 1, 2, 0, 1, 3, 0, 1, 3, 1, 0, 1, 4, 3, 0, 1, 4, 4, 0, 1, 5, 6, 0, 1, 5, 7, 0, 1, 6, 9, 1, 0, 1, 6, 11, 4, 0, 1, 7, 13, 7, 0, 1, 7, 15, 11, 0, 1, 8, 18, 15, 0, 1, 8, 20, 19, 0, 1, 9, 23, 25, 1, 0, 1, 9, 26, 30, 5
OFFSET
0,14
LINKS
EXAMPLE
T(10,1) = 1: [10].
T(10,2) = 4: [5,5], [4,6], [3,7], [2,8].
T(10,3) = 3: [3,3,4], [2,4,4], [2,3,5].
Triangle T(n,k) begins:
1;
0;
0, 1;
0, 1;
0, 1;
0, 1, 1;
0, 1, 2;
0, 1, 2;
0, 1, 3;
0, 1, 3, 1;
0, 1, 4, 3;
0, 1, 4, 4;
0, 1, 5, 6;
0, 1, 5, 7;
0, 1, 6, 9, 1;
...
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, (p-> expand(
x*(p-coeff(p, x, i-1)*x^(i-1))))(b(n-i, i)))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..max(0, degree(p))))(b(n$2)):
seq(T(n), n=0..30);
MATHEMATICA
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, Function[p, Expand[x*(p - Coefficient[p, x, i-1]*x^(i-1))]][b[n-i, i]]]] ]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Max[0, Exponent[p, x]]}]][b[n, n]]; Table[T[n], {n, 0, 30}] // Flatten (* Jean-François Alcover, Feb 08 2017, translated from Maple *)
CROSSREFS
Columns k=0-10 give: A000007, A000012 (for n>1), A004526(n-2) (for n>4), A244239, A244240, A244241, A244242, A244243, A244244, A244245, A244246.
Row sums give A238394.
Cf. A052146.
Sequence in context: A163325 A105186 A328346 * A058709 A025842 A141100
KEYWORD
nonn,tabf,look
AUTHOR
Alois P. Heinz, Feb 26 2014
STATUS
approved