0,3

Column 0 forms A082161. Column 1 forms A102087. Row sums form A102088.

T(n, 0) = A082161(n) for n>0, with T(0, 0) = 1.

G.f. for column k: T(k, k) = k+1 = Sum_{n>=0} T(n+k, k)*x^n*prod_{j=1, n+1} (1-(j+k)*x).

Rows of T begin:

[1],

[1,2],

[3,4,3],

[16,20,9,4],

[127,156,63,16,5],

[1363,1664,648,144,25,6],

[18628,22684,8703,1840,275,36,7],

[311250,378572,144243,29824,4200,468,49,8],

[6173791,7504640,2849400,582640,79775,8316,735,64,9],...

Matrix square T^2 equals T excluding the main diagonal:

[1],

[3,4],

[16,20,9],

[127,156,63,16],

[1363,1664,648,144,25],...

G.f. for column 0: 1 = (1-x) + 1*x*(1-x)(1-2x) + 3*x^2*(1-x)(1-2x)(1-3x) + ... + T(n,0)*x^n*(1-x)(1-2x)(1-3x)*..*(1-(n+1)*x) + ...

G.f. for column 1: 2 = 2(1-2x) + 4*x*(1-2x)(1-3x) + 20*x^2*(1-2x)(1-3x)(1-4x) + ... + T(n+1,1)*x^n*(1-2x)(1-3x)(1-4x)*..*(1-(n+2)*x) + ...

G.f. for column 2: 3 = 3(1-3x) + 9*x*(1-3x)(1-4x) + 63*x^2*(1-3x)(1-4x)(1-5x) + ... + T(n+2,2)*x^n*(1-3x)(1-4x)(1-5x)*..*(1-(n+3)*x) + ...

{T(n, k)=local(A=matrix(1, 1), B); A[1, 1]=1; for(m=2, n+1, B=matrix(m, m); for(i=1, m, for(j=1, i, if(j==i, B[i, j]=j, if(j==1, B[i, j]=(A^2)[i-1, 1], B[i, j]=(A^2)[i-1, j])); )); A=B); return(A[n+1, k+1])}

(PARI) {T(n, k)=if(n<k, 0, if(n==k, k+1, polcoeff(1-sum(i=k, n-1, T(i, k)*x^i*prod(j=1, i-k+1, 1-(j+k)*x+x*O(x^n))), n)))}

Cf. A082161, A102087, A102088.

Cf. A102316.

Adjacent sequences: A102083 A102084 A102085 this_sequence A102087 A102088 A102089

Sequence in context: A098596 A058267 A048259 this_sequence A009184 A009716 A009590

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 29 2004