OFFSET
0,5
COMMENTS
Array read by antidiagonals: (0,1),(0,2),(1,1),(0,3),...
The first column is A082157.
LINKS
G. C. Greubel, Antidiagonals n = 0..50, flattened
Valery A. Liskovets, Exact enumeration of acyclic automata, Proc. 15th Conf. "Formal Power Series and Algebr. Combin. (FPSAC'03)", 2003.
Valery A. Liskovets, Exact enumeration of acyclic deterministic automata, Discrete Appl. Math., 154, No.3 (2006), 537-551.
FORMULA
T(n, k) = T_2(n, k) where T_2(0, k) = 1, T_2(n, k) = Sum_{i=0..n-1} (-1)^(n-i-1)*binomial(n, i)*(i+k)^(2*(n-i))*T_2(i, k), n > 0.
EXAMPLE
The array begins:
1, 1, 1, 1, 1, ...;
1, 4, 9, 16, 25, ...;
7, 56, 207, 544, 1175, ...;
142, 1780, 9342, 32848, 91150, ...;
5941, 103392, 709893, 3142528, 10682325, ...;
428856, 9649124, 82305144, 440535696, 1775027000, ...;
47885899, 1329514816, 13598786979, 85529171200, ...;
Antidiagonal triangle begins as:
1;
1, 1;
1, 4, 7;
1, 9, 56, 142;
1, 16, 207, 1780, 5941;
1, 25, 544, 9342, 103392, 428856;
1, 36, 1175, 32848, 709893, 9649124, 47885899;
1, 49, 2232, 91150, 3142528, 82305144, 1329514816, 7685040448;
MATHEMATICA
T[0, _]= 1; T[n_, k_]:= T[n, k]= Sum[Binomial[n, i] (-1)^(n-i-1)*(i + k)^(2n-2i) T[i, k], {i, 0, n-1}];
Table[T[n-k-1, k], {n, 1, 10}, {k, n-1, 1, -1}]//Flatten (* Jean-François Alcover, Aug 29 2019 *)
PROG
(Magma)
function A(n, k)
if n eq 0 then return 1;
else return (&+[(-1)^(n-j+1)*Binomial(n, j)*(k+j)^(2*n-2*j)*A(j, k): j in [0..n-1]]);
end if;
end function;
A082169:= func< n, k | A(k, n-k+1) >;
[A082169(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jan 19 2024
(SageMath)
@CachedFunction
def A(n, k):
if n==0: return 1
else: return sum((-1)^(n-j+1)*binomial(n, j)*(k+j)^(2*n-2*j)*A(j, k) for j in range(n))
def A082169(n, k): return A(k, n-k+1)
flatten([[A082169(n, k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Jan 19 2024
CROSSREFS
KEYWORD
AUTHOR
Valery A. Liskovets, Apr 09 2003
STATUS
approved