OFFSET
0,8
COMMENTS
This number triangle satisfies T(n,k) = A008284(n+k+1,2*k+1), n,k >= 0. This means that T(n,k) is also the number of partitions of N:=n+k+1 into M:=2*k+1 parts. For the proof add 1 to every odd part of each partition of n':=2*n+1 into m:=2*k+1 parts which are all odd, and divide each part by a factor of 2, thus obtaining a partition of n+k+1 into m=2*k+1 parts. All partitions of N,for N>=1, into an odd number of parts M (M from {1,...,N}) are reached: just take k=(M-1)/2 and n=N-1-k. Each partition of N into an odd number of parts can only arise once from the given recipe (for given N and M the k and n values are unique). See also a comment by Franklin T. Adams-Watters on A152140. - Wolfdieter Lang, Jul 09 2012
LINKS
Alois P. Heinz, Rows n = 0..200, flattened
FORMULA
T(n,k) = A152140(2n+1,2k+1).
T(n,k) = p(n+k+1,2*k+1), n >= 0, k >= 0, with p(N,M)= A008284(N,M), the number of partitions of N into M parts. See the sketch of the proof given above as a comment. - Wolfdieter Lang, Jul 09 2012
O.g.f. for column k: (x^k)/product(1-x^j,j=1..(2*k+1)), k>=0.
From the o.g.f.s of A008284. - Wolfdieter Lang, Jul 10 2012
EXAMPLE
Triangle begins:
1
1 1
1 1 1
1 2 1 1
1 3 2 1 1
1 4 3 2 1 1
1 5 5 3 2 1 1
1 7 7 5 3 2 1 1
1 8 10 7 5 3 2 1 1
1 10 13 11 7 5 3 2 1 1
1 12 18 15 11 7 5 3 2 1 1
1 14 23 21 15 11 7 5 3 2 1 1
1 16 30 28 22 15 11 7 5 3 2 1 1
1 19 37 38 30 22 15 11 7 5 3 2 1 1
1 21 47 49 41 30 22 15 11 7 5 3 2 1 1
1 24 57 65 54 42 30 22 15 11 7 5 3 2 1 1
1 27 70 82 73 56 42 30 22 15 11 7 5 3 2 1 1
1 30 84 105 94 76 56 42 30 22 15 11 7 5 3 2 1 1
1 33 101 131 123 99 77 56 42 30 22 15 11 7 5 3 2 1 1
1 37 119 164 157 131 101 77 56 42 30 22 15 11 7 5 3 2 1 1
From Wolfdieter Lang, Jul 09 2012 (Start)
T(5,1) = 4 from the four partitions of 11 into 3 parts, all of which are odd: [1,1,9], [1,3,7], [1,5,5] and [3,3,5].
T(5,1) = 4 from the four partitions of 7 = 5+1+1 into 3 parts:
[1,1,5], [1,2,4], [1,3,3] and [2,2,3].
(End)
MAPLE
b:= proc(n, i) option remember; `if`(n=0, 1/sqrt(x), `if`(i<1, 0,
b(n, i-2)+`if`(i>n, 0, expand(sqrt(x)*b(n-i, i)))))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(2*n+1, 2*n+1)):
seq(T(n), n=0..12); # Alois P. Heinz, Jun 21 2021
MATHEMATICA
(* p = A008284 *) p[n_, 1] = 1; p[n_, k_] := p[n, k] = If[n >= k, Sum[p[n - i, k - 1], {i, 1, n - 1}] - Sum[p[n - i, k], {i, 1, k - 1}], 0];
T[n_, k_] := p[n + k + 1, 2 k + 1];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Oct 28 2019, after Wolfdieter Lang *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
R. J. Mathar, Sep 25 2009
EXTENSIONS
Indices corrected by R. J. Mathar, Jul 09 2012
STATUS
approved