A082160
Deterministic completely defined acyclic automata with 3 inputs and n+1 transient labeled states including a unique state having all transitions to the absorbing state.
Original entry on oeis.org
1, 7, 315, 45682, 15646589, 10567689552, 12503979423607, 23841011541867520, 68835375121428936153, 286850872894190847235840, 1660638682341609286358474579, 12947089879912710544534553836032
Offset: 0
- G. C. Greubel, Table of n, a(n) for n = 0..150
- 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.
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function a(n) // a = A082160
if n eq 0 then return 1;
else return (&+[Binomial(n,j)*(-1)^(n-j-1)*((j+2)^3 - 1)^(n-j)*a(j): j in [0..n-1]]);
end if;
end function;
[a(n): n in [0..20]]; // G. C. Greubel, Jan 17 2024
-
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n, i] (-1)^(n - i - 1) ((i + 2)^3 - 1)^(n - i) a[i], {i, 0, n - 1}];
Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Aug 29 2019 *)
-
@CachedFunction
def a(n): # A082160
if n==0: return 1
else: return sum(binomial(n,j)*(-1)^(n-j-1)*((j+2)^3 -1)^(n-j)*a(j) for j in range(n))
[a(n) for n in range(21)] # G. C. Greubel, Jan 17 2024
A082164
Deterministic completely defined initially connected acyclic automata with 3 inputs and n+1 transient unlabeled states including a unique state having all transitions to the absorbing state.
Original entry on oeis.org
1, 7, 133, 5362, 380093, 42258384, 6830081860, 1520132414241, 447309239576913, 168599289097947589, 79364534944804317166, 45701029702436877135199, 31642128418550547009710906, 25960688434777959685891570936, 24926392120419324125117256758595, 27708074645788511889179577045508824
Offset: 1
-
b[, 0, ] = 1; b[k_, n_, r_] := b[k, n, r] = Sum[Binomial[n, t] (-1)^(n - t - 1) ((t + r + 1)^k - 1)^(n - t) b[k, t, r], {t, 0, n - 1}];
d3[n_] := d3[n] = b[3, n, 1] - Sum[Binomial[n - 1, j - 1] T3[n - j, j + 1] d3[j], {j, 1, n - 1}];
T3[0, ] = 1; T3[n, k_] := T3[n, k] = Sum[Binomial[n, i] (-1)^(n - i - 1) ((i + k + 1)^3 - 1)^(n - i) T3[i, k], {i, 0, n - 1}];
a[n_] := If[n == 1, 1, d3[n - 1]/(n - 2)!];
Array[a, 20] (* Jean-François Alcover, Aug 29 2019 *)
A103243
Unreduced numerators of the elements T(n,k)/(n-k)!, read by rows, of the triangular matrix P^-1, which is the inverse of the matrix defined by P(n,k) = (1-(k+1)^3)^(n-k)/(n-k)! for n >= k >= 1.
Original entry on oeis.org
1, 7, 1, 315, 26, 1, 45682, 2600, 63, 1, 15646589, 675194, 11655, 124, 1, 10567689552, 366349152, 4861458, 37944, 215, 1, 12503979423607, 361884843866, 3882676581, 23641468, 100835, 342, 1, 23841011541867520, 591934698991168, 5318920238688
Offset: 1
Rows of unreduced fractions T(n,k)/(n-k)! begin:
[1/0! ],
[7/1!, 1/0! ],
[315/2!, 26/1!, 1/0! ],
[45682/3!, 2600/2!, 63/1!, 1/0! ],
[15646589/4!, 675194/3!, 11655/2!, 124/1!, 1/0! ],
[10567689552/5!, 366349152/4!, 4861458/3!, 37944/2!, 215/1!, 1/0! ], ...
forming the inverse of matrix P where P(n,k) = A103247(n,k)/(n-k)!:
[1/0! ],
[ -7/1!, 1/0! ],
[49/2!, -26/1!, 1/0! ],
[ -343/3!, 676/2!, -63/1!, 1/0! ],
[2401/4!, -17576/3!, 3969/2!, -124/1!, 1/0! ],
[ -16807/5!, 456976/4!, -250047/3!, 15376/2!, -215/1!, 1/0! ], ...
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{T(n,k)=my(P);if(n>=k&k>=1, P=matrix(n,n,r,c,if(r>=c,(1-(c+1)^3)^(r-c)/(r-c)!))); return(if(n
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