cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Valery A. Liskovets, Apr 09 2003

Keywords

Comments

This is the first column of the array A082172.

Crossrefs

Programs

  • Magma
    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
    
  • Mathematica
    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 *)
  • SageMath
    @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

Formula

a(n) = b_3(n) where b_3(0) = 1, b_3(n) = Sum_{i=0..n-1} binomial(n, i)*(-1)^(n-i-1)*((i+2)^3 - 1)^(n-i)*b_3(i), n > 0.

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

Views

Author

Valery A. Liskovets, Apr 09 2003

Keywords

Comments

Coefficients T_2(n,k) form the array A082172. These automata have no nontrivial automorphisms (by states).

Crossrefs

Programs

  • Mathematica
    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 *)

Formula

a(n) := d_3(n)/(n-1)! where d_3(n) := b_3(n, 1)-sum(binomial(n-1, j-1)*T_3(n-j, j+1)*d_3(j), j=1..n-1); and T_3(0, k) := 1, T_3(n, k) := sum(binomial(n, i)*(-1)^(n-i-1)*((i+k+1)^3-1)^(n-i)*T_3(i, k), i=0..n-1), n>0.

Extensions

More terms from 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

Views

Author

Paul D. Hanna, Feb 02 2005

Keywords

Comments

Define triangular matrix P by P(n,k) = (-k^3-3k^2-3k)^(n-k)/(n-k)!, then M = P*D*P^-1 = A103237 satisfies: M^3 + 3M^2 + 3M = SHIFTUP(M) where D is the diagonal matrix consisting of {1,2,3,...}. The operation SHIFTUP(M) shifts each column of M up 1 row. Essentially equal to square array A082172 as a triangular matrix. The first column is A082160 (quasi-acyclic automata with 3 inputs).

Examples

			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! ], ...
		

Crossrefs

Programs

  • PARI
    {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
    				

Formula

For n > k >= 1: 0 = Sum_{m=k..n} C(n-k, m-k)*(1-(m+1)^3)^(n-m)*T(m, k). For n > k >= 1: 0 = Sum_{j=k..n} C(n-k, j-k)*(1-(k+1)^3)^(j-k)*T(n, j).
Showing 1-3 of 3 results.