A345123 Number T(n,k) of ordered subsequences of {1,...,n} containing at least k elements and such that the first differences contain only odd numbers; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
1, 2, 1, 4, 3, 1, 7, 6, 3, 1, 12, 11, 7, 3, 1, 20, 19, 14, 8, 3, 1, 33, 32, 26, 17, 9, 3, 1, 54, 53, 46, 34, 20, 10, 3, 1, 88, 87, 79, 63, 43, 23, 11, 3, 1, 143, 142, 133, 113, 83, 53, 26, 12, 3, 1, 232, 231, 221, 196, 156, 106, 64, 29, 13, 3, 1, 376, 375, 364, 334, 279, 209, 132, 76, 32, 14, 3, 1
Offset: 0
Examples
T(0,0) = 1: [].
T(1,1) = 1: [1].
T(2,2) = 1: [1,2].
T(3,1) = 6: [1], [2], [3], [1,2], [2,3], [1,2,3].
T(4,0) = 12: [], [1], [2], [3], [4], [1,2], [1,4], [2,3], [3,4], [1,2,3], [2,3,4], [1,2,3,4].
T(6,3) = 17: [1,2,3], [1,2,5], [1,4,5], [2,3,4], [2,3,6], [2,5,6], [3,4,5], [4,5,6], [1,2,3,4], [1,2,3,6], [1,2,5,6], [1,4,5,6], [2,3,4,5], [3,4,5,6], [1,2,3,4,5], [2,3,4,5,6], [1,2,3,4,5,6].
Triangle T(n,k) begins:
1;
2, 1;
4, 3, 1;
7, 6, 3, 1;
12, 11, 7, 3, 1;
20, 19, 14, 8, 3, 1;
33, 32, 26, 17, 9, 3, 1;
54, 53, 46, 34, 20, 10, 3, 1;
88, 87, 79, 63, 43, 23, 11, 3, 1;
143, 142, 133, 113, 83, 53, 26, 12, 3, 1;
232, 231, 221, 196, 156, 106, 64, 29, 13, 3, 1;
...
References
- Chu, Hung Viet, Various Sequences from Counting Subsets, Fib. Quart., 59:2 (May 2021), 150-157.
Links
- Alois P. Heinz, Rows n = 0..140, flattened
- Chu, Hung Viet, Various Sequences from Counting Subsets, arXiv:2005.10081 [math.CO], 2021.
Crossrefs
Programs
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Maple
b:= proc(n, l, t) option remember; `if`(n=0, `if`(t=0, 1, 0), `if`(0 in [l, irem(1+l-n, 2)], b(n-1, n, max(0, t-1)), 0)+b(n-1, l, t)) end: T:= (n, k)-> b(n, 0, k): seq(seq(T(n, k), k=0..n), n=0..10); # second Maple program: g:= proc(n, k) option remember; `if`(k>n, 0, `if`(k in [0, 1], n^k, g(n-1, k-1)+g(n-2, k))) end: T:= proc(n, k) option remember; `if`(k>n, 0, g(n, k)+T(n, k+1)) end: seq(seq(T(n, k), k=0..n), n=0..10); # third Maple program: T:= proc(n, k) option remember; `if`(k>n, 0, binomial(iquo(n+k, 2), k)+ `if`(k>0, binomial(iquo(n+k-1, 2), k), 0)+T(n, k+1)) end: seq(seq(T(n, k), k=0..n), n=0..10); -
Mathematica
T[n_, k_] := T[n, k] = If[k > n, 0, Binomial[Quotient[n+k, 2], k] + If[k > 0, Binomial[Quotient[n+k-1, 2], k], 0] + T[n, k+1]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 11}] // Flatten (* Jean-François Alcover, Nov 06 2021, after 3rd Maple program *)
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