A103238 -id:A103238 - cp's OEIS frontend

A103239 Column 0 of triangular matrix T = A103238, which satisfies: T^2 + T = SHIFTUP(T) where diagonal(T)={1,2,3,...}.

1, 2, 8, 52, 480, 5816, 87936, 1601728, 34251520, 843099616, 23520367488, 734404134336, 25402332040704, 964965390917120, 39964015456707584, 1793140743838290432, 86691698782589288448, 4494521175128812273152

Offset: 0

Paul D. Hanna, Jan 31 2005

nonn

a(n-1) = number of initially connected acyclic unlabeled n-state automata on a 2-letter input alphabet for which only one state is affected identically by both input letters. This state is necessarily one that is carried to the sink (absorbing state). For example, with n=2, a(1)=2 counts 2333, 3233, but not 2233. Here 1 is the source and 3 is the sink and 2333 is short for {{1, 2}, {1, 3}, {2, 3}, {2, 3}} giving the action of the input letters. The unlabeled condition is captured by requiring that the first appearances of 2,3,...,n occur in that order. A082161 counts these automata without the affected-identically restriction. - David Callan, Jun 07 2006

			1 = (1-2x) + 2*x/(1-x)*(1-2x)(1-3x) + 8*x^2/(1-x)^2*(1-2x)(1-3x)(1-4x) +
52*x^3/(1-x)^3*(1-2x)(1-3x)(1-4x)(1-5x) + ...
+ a(n)*x^n/(1-x)^n*(1-2x)(1-3x)*..*(1-(n+2)x) + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..32

Cf. A103238.

{a(n)=if(n<0,0,if(n==0,1,polcoeff( 1-sum(k=0,n-1,a(k)*x^k/(1-x)^k*prod(j=0,k,1-(j+2)*x+x*O(x^n))),n)))}

G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1-x)^n*Product_{j=0..n} (1-(j+2)*x).

A103244 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) = (-k^2-k)^(n-k)/(n-k)! for n >= k >= 1.

Original entry on oeis.org

1, 2, 1, 20, 6, 1, 512, 108, 12, 1, 25392, 4104, 336, 20, 1, 2093472, 273456, 17568, 800, 30, 1, 260555392, 28515456, 1500288, 54800, 1620, 42, 1, 45819233280, 4311418752, 191549952, 5808000, 140400, 2940, 56, 1, 10849051434240, 894918533760, 34352605440, 887256000, 18033840, 313992, 4928, 72, 1

Offset: 1

Paul D. Hanna, Feb 02 2005

Define triangular matrix P by P(n,k) = (-k^2-k)^(n-k)/(n-k)!, then M = P*D*P^-1 = A103238 satisfies: M^2 + M = 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.

First column is A103353.

			This triangle begins:
1;
2, 1;
20, 6, 1;
512, 108, 12, 1;
25392, 4104, 336, 20, 1;
2093472, 273456, 17568, 800, 30, 1;
260555392, 28515456, 1500288, 54800, 1620, 42, 1;
45819233280, 4311418752, 191549952, 5808000, 140400, 2940, 56, 1;
10849051434240, 894918533760, 34352605440, 887256000, 18033840, 313992, 4928, 72, 1; ...
Rows of unreduced fractions T(n,k)/(n-k)! begin:
[1/0!],
[2/1!, 1/0!],
[20/2!, 6/1!, 1/0!],
[512/3!, 108/2!, 12/1!, 1/0!],
[25392/4!, 4104/3!, 336/2!, 20/1!, 1/0!],
[2093472/5!, 273456/4!, 17568/3!, 800/2!, 30/1!, 1/0!],...
forming the inverse of matrix P where P(n,k) = (-1)^(n-k)*(k^2+k)^(n-k)/(n-k)!:
[1/0!],
[ -2/1!, 1/0!],
[4/2!, -6/1!, 1/0!],
[ -8/3!, 36/2!, -12/1!, 1/0!],
[16/4!, -216/3!, 144/2!, -20/1!, 1/0!], ...

Cf. A261642, A103238, A103353.

nmax = 9;
P = Table[If[n >= k, (-k^2-k)^(n-k)/(n-k)!, 0], {n, 1, nmax}, {k, 1, nmax}] // Inverse;
T[n_, k_] := If[n < k || k < 1, 0, P[[n, k]]*(n - k)!];
Table[T[n, k], {n, 1, nmax}, {k, 1, n}] // Flatten (* Jean-François Alcover, Aug 09 2018, from PARI *)

{T(n,k)=local(P);if(n>=k&k>=1, P=matrix(n,n,r,c,if(r>=c,(-c^2-c)^(r-c)/(r-c)!))); return(if(n

For n > k >= 1: 0 = Sum_{m=k..n} C(n-k, m-k)*(-m^2-m)^(n-m)*T(m, k).

For n > k >= 1: 0 = Sum_{j=k..n} C(n-k, j-k)*(-k^2-k)^(j-k)*T(n, j).

cp's OEIS Frontend

A103239 Column 0 of triangular matrix T = A103238, which satisfies: T^2 + T = SHIFTUP(T) where diagonal(T)={1,2,3,...}.

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