OFFSET
0,3
COMMENTS
Array read by descending antidiagonals: (0,1), (0,2), (1,1), (0,3), ...
The first column is A082159; i.e., T(n,k=1) = A082159(n). [The number n of transient states in the name of square array T(n,k) does not include the pre-dead transient state, which is, however, included in the name of A082159. See Section 3.1 in Liskovets (2006). - Petros Hadjicostas, Mar 07 2021]
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) = S_2(n, k) where S_2(0, k) := 1 and S_2(n, k) := Sum_{i=0..n-1} binomial(n, i)*(-1)^(n-i-1)*((i + k + 1)^2 - 1)^(n-i)*S_2(i, k) for n > 0.
EXAMPLE
Array T(n,k) (with rows n >= 0 and columns k >= 1) begins:
1, 1, 1, 1, 1, ...;
3, 8, 15, 24, 35, ...;
39, 176, 495, 1104, 2135, ...;
1206, 7784, 29430, 84600, 204470, ...;
69189, 585408, 2791125, 9841728, 28569765, ...;
6416568, 67481928, 389244600, 1627740504, ...;
881032059, 11111547520, 75325337235, ...;
...
Triangular array A(n,k) = T(k-1, n-k+1) (with rows n >= 1 and columns k = 1..n), read from the antidiagonals downwards of square array T:
1;
1, 3,
1, 8, 39;
1, 15, 176, 1206;
1, 24, 495, 7784, 69189;
1, 35, 1104, 29430, 585408, 6416568;
1, 48, 2135, 84600, 2791125, 67481928, 881032059;
...
MATHEMATICA
T[0, _] = 1; T[n_, k_] := T[n, k] = Sum[Binomial[n, i] (-1)^(n - i - 1)*((i + k + 1)^2 - 1)^(n - i)*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
(PARI) lista(nn, kk)={my(T=matrix(nn+1, kk)); for(n=1, nn+1, for(k=1, kk, T[n, k] = if(n==1, 1, sum(i=0, n-2, binomial(n-1, i)*(-1)^(n-i-2)*((i + k + 1)^2 - 1)^(n-i-1)*T[i+1, k])))); T; } \\ Petros Hadjicostas, Mar 07 2021
(Magma)
function A(n, k)
if n eq 0 then return 1;
else return (&+[(-1)^(n-j+1)*Binomial(n, j)*((k+j+1)^2-1)^(n-j)*A(j, k): j in [0..n-1]]);
end if;
end function;
A082171:= func< n, k | A(k, n-k+1) >;
[A082171(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+1)^2-1)^(n-j)*A(j, k) for j in range(n))
def A082171(n, k): return A(k, n-k+1)
flatten([[A082171(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
EXTENSIONS
Name clarified by Petros Hadjicostas, Mar 07 2021
STATUS
approved