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A027356
Array read by rows: T(n,k) = number of partitions of n into distinct odd parts in which k is the greatest part, for k=1,2,...,n, n>=1.
6
1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0
OFFSET
1,145
COMMENTS
First T(n,k) not 0 or 1 is T(17,9)=2, which counts 1+7+9 and 3+5+9. Row sums: A000700.
LINKS
Sean A. Irvine, Java program (github)
FORMULA
T(n, 1)=0 for all n; T(n, n)=1 for all odd n>1; and for n>=3, T(n, k)=0 if k is even, else T(n, k)=Sum{T(n-k, i): i=1, 2, ..., n-1} for k=2, 3, ..., n-1.
EXAMPLE
First 5 rows:
1
0 0
0 0 1
0 0 1 0
0 0 0 0 1
Row 40 with even-numbered terms deleted:
0 0 0 0 0 0 2 5 6 7 6 5 4 3 2 1 1 1 1;
E.g. final 2 counts these two partitions: 9+31 and 1+3+5+31.
MAPLE
b:= proc(n, i) option remember; `if`(n>i^2, 0, `if`(n=0, 1,
b(n, i-1) +(p-> `if`(p>n, 0, b(n-p, i-1)))((2*i-1))))
end:
T:= (n, k)-> `if`(k::even, 0, b(n-k, (k-1)/2)):
seq(seq(T(n, k), k=1..n), n=1..20); # Alois P. Heinz, Oct 28 2019
MATHEMATICA
b[n_, i_] := b[n, i] = If[n > i^2, 0, If[n == 0, 1, b[n, i - 1] + Function[p, If[p > n, 0, b[n - p, i - 1]]][2i - 1]]];
T [n_, k_] := If[EvenQ[k], 0, b[n - k, (k - 1)/2]];
Table[Table[T[n, k], {k, 1, n}], {n, 1, 20}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)
CROSSREFS
Cf. A000700.
T(4n+1,2n+1) gives A069910.
Sequence in context: A173856 A288926 A373840 * A361114 A181663 A359370
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, revised Jul 23 2004
EXTENSIONS
Edited by N. J. A. Sloane, Sep 14 2008 at the suggestion of R. J. Mathar
STATUS
approved