0,4

Column 0 equals A122455 if we define A122455(0)=1.

T(n,k) = [x^(n-k)] Sum_{j=0..n} C(n,j)*x^j/(1-j*x)^k /[Product_{i=0..j}(1-i*x)].

Triangle T begins:

1;

1, 1;

3, 2, 1;

13, 9, 3, 1;

71, 46, 18, 4, 1;

456, 285, 110, 30, 5, 1;

3337, 2021, 780, 215, 45, 6, 1;

27203, 16023, 6167, 1729, 371, 63, 7, 1;

243203, 139812, 53494, 15176, 3346, 588, 84, 8, 1;

2357356, 1326111, 504030, 143814, 32376, 5886, 876, 108, 9, 1; ...

Let P denote the matrix equal to Pascal's triangle shift down 1 row:

P(n,k) = C(n+1,k) for n>k>=0, with P(n,n)=1 for n>=0.

Illustrate row n of T = row n of P^n as follows.

Matrix P = I + D*C begins:

1;

1, 1;

1, 1, 1;

1, 2, 1, 1;

1, 3, 3, 1, 1;

1, 4, 6, 4, 1, 1; ...

Matrix cube P^3 begins:

1;

3, 1;

6, 3, 1;

13, 9, 3, 1; <== row 3 of P^3 = row 3 of T

30, 25, 12, 3, 1;

73, 72, 40, 15, 3, 1; ...

Matrix 4-th power P^4 begins:

1;

4, 1;

10, 4, 1;

26, 14, 4, 1;

71, 46, 18, 4, 1; <== row 4 of P^4 = row 4 of T

204, 155, 70, 22, 4, 1; ...

Matrix 5-th power P^5 begins:

1;

5, 1;

15, 5, 1;

45, 20, 5, 1;

140, 75, 25, 5, 1;

456, 285, 110, 30, 5, 1; <== row 5 of P^5 = row 5 of T.

(PARI) /* As generated by the g.f.: */ {T(n, k)=polcoeff(sum(j=0, n, binomial(n, j)*x^j/(1-j*x)^k/prod(i=0, j, 1-i*x+x*O(x^(n-k)))), n-k)} /* As generated by matrix power: row n of T equals row n of P^n: */ {T(n, k)=local(P=matrix(n+1, n+1, r, c, if(r==c, 1, if(r>c, binomial(r-2, c-1))))); (P^n)[n+1, k+1]}

Cf. columns:, A134091, A134092, A134093; A134094 (row sums).

Sequence in context: A048647 A059438 A104980 this_sequence A132845 A129652 A127126

Adjacent sequences: A134087 A134088 A134089 this_sequence A134091 A134092 A134093

nonn,tabl

Paul D. Hanna (pauldhanna(AT)juno.com), Oct 07 2007