0,7

Amazingly, A126460 = A126445^-1*A126450 = A126450^-1*A126454 = A126454^-1*A126457; and also A126465 = A126450*A126445^-1 = A126454*A126450^-1 = A126457*A126454^-1. Also, column k equals unsigned column k of the matrix inverse of triangle P_k defined by P_k(m,j) = C( C(j+2,3) - C(k+2,3) + m-j, m-j) for m>=j>=0.

G.f. of column k: 1/(1-x) = Sum_{n>=0} T(n+k,k)*x^n*(1-x)^p_k(n), so that column k equals the number of subpartitions of the partition p_k defined by: p_k(n) = (n^2 + (3*k+3)*n + (3*k^2+6*k-4))*n/6 for n>=0.

Triangle T begins:

1;

1, 1;

1, 1, 1;

3, 3, 1, 1;

21, 21, 6, 1, 1;

274, 274, 75, 10, 1, 1;

5806, 5806, 1565, 195, 15, 1, 1;

182766, 182766, 48950, 5940, 420, 21, 1, 1;

8034916, 8034916, 2145626, 257300, 17570, 798, 28, 1, 1; ...

where column 1 of T^1 equals left-shifted column 0 of T.

Matrix cube T^3 begins:

1;

3, 1;

6, 3, (1);

22, 12, (3), 1;

163, 91, (21), 3, 1;

2167, 1219, (274), 33, 3, 1;

46248, 26091, (5806), 661, 48, 3, 1;

1460301, 824853, (182766), 20341, 1369, 66, 3, 1; ...

where column 2 of T^3 equals left-shifted column 1 of T.

Matrix power T^6 begins:

1;

6, 1;

21, 6, 1;

98, 33, 6, (1);

791, 281, 51, (6), 1;

10850, 3929, 710, (75), 6, 1;

234472, 85557, 15425, (1565), 105, 6, 1;

7444172, 2725402, 490806, (48950), 3080, 141, 6, 1; ...

where column 3 of T^6 equals left-shifted column 2 of T.

(PARI) {T(n, k)=abs((matrix(n+1, n+1, r, c, binomial((c-1)*c*(c+1)/3!-k*(k+1)*(k+2)/3!+r-c, r-c))^-1)[n+1, k+1])} (PARI) /* As Defined by Matrix Product A126460 = A126445^-1*A126450: */ {T(n, k)=local(M=matrix(n+1, n+1, r, c, if(r>=c, binomial((r-1)*r*(r+1)/3!-(c-1)*c*(c+1)/3!, r-c))), N=matrix(n+1, n+1, r, c, if(r>=c, binomial((r-1)*r*(r+1)/3!-(c-1)*c*(c+1)/3!+1, r-c)))); (M^-1*N)[n+1, k+1]}

Columns: A126461, A126462, A126463, A126464; A126465 (dual); A107876 (variant); subpartitions defined: A115728.

Sequence in context: A075837 A087107 A155170 this_sequence A100940 A063421 A073067

Adjacent sequences: A126457 A126458 A126459 this_sequence A126461 A126462 A126463

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Dec 27 2006