OFFSET
0,19
COMMENTS
Conjecture: Column k > 0 has a linear recurrence (with constant coefficients) of order = A005317(k) = (2^k + C(2*k,k))/2. - Vaclav Kotesovec, Jul 07 2015
From Vaclav Kotesovec, Jul 07 2015: (Start) For k > 1, A(n,k) ~ c(k) * d(k)^n
k c(k) d(k)
2 0.2840509026895102746628049030651... 1.8832035059135258641689474653620...
3 0.1678494211968692989590951622212... 2.6304414743928951523517253855770...
4 0.0973070675347403976445165510589... 3.3758288741377846847522960161445...
5 0.0552389982575367440330445172521... 4.1183824671958029895499633437571...
6 0.0309726120341077011398575643793... 4.8588208495640240252838055706997...
7 0.0172064353582683268003622374813... 5.5979905586951369718393573797927...
8 0.0094902135663231445267663712259... 6.3363450921766600853069060904417...
9 0.00520430877801650454166967632... 7.0741444217884608367707985...
10 0.0028405987031922... 7.811548995086...
(End)
LINKS
Alois P. Heinz, Antidiagonals n = 0..36, flattened
FORMULA
A(n,k) = Sum_{j=0..k} A259784(n,j).
EXAMPLE
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, ...
0, 0, 0, 0, 0, 0, 0, 0, ...
0, 1, 1, 1, 1, 1, 1, 1, ...
0, 0, 2, 2, 2, 2, 2, 2, ...
0, 1, 4, 9, 9, 9, 9, 9, ...
0, 0, 6, 24, 44, 44, 44, 44, ...
0, 1, 13, 57, 168, 265, 265, 265, ...
0, 0, 24, 140, 536, 1280, 1854, 1854, ...
MAPLE
b:= proc(n, s, k) option remember; `if`(n=0, 1, `if`(n+k in s,
b(n-1, (s minus {n+k}) union `if`(n-k>1, {n-k-1}, {}), k),
add(`if`(j=n, 0, b(n-1, (s minus {j}) union
`if`(n-k>1, {n-k-1}, {}), k)), j=s)))
end:
A:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), b(n, {$max(1, n-k)..n}, k)):
seq(seq(A(n, d-n), n=0..d), d=0..12);
MATHEMATICA
b[n_, s_, k_] := b[n, s, k] = If[n==0, 1, If[MemberQ[s, n+k], b[n-1, Join[s ~Complement~ {n+k}] ~Union~ If[n-k>1, {n-k-1}, {}], k], Sum[If[j==n, 0, b[n -1, Join[s ~Complement~ {j}] ~Union~ If[n-k>1, {n-k-1}, {}], k]], {j, s}]] ];
A[n_, k_] := If[k == 0, If[n == 0, 1, 0], b[n, Range[Max[1, n-k], n], k]];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Mar 29 2017, translated from Maple *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 05 2015
STATUS
approved