0,2

In the language of the Shapiro et al. reference (given in A053121) such a lower triangular (ordinary) convolution array, considered as a matrix, belongs to the Bell-subgroup of the Riordan-group. The G.f. for the row polynomials p(n,x) (increasing powers of x) is 1/(sqrt(1-4*z)-x*z).

The column sequences are for m=0..20: A000984, A000302 (powers of 4), A002457, A002697, A002802, A038845, A020918, A038846, A020920, A040075, A020922, A045543, A020924, A054337, A020926, A054338, A020928, A054339, A020930, A054340, A020932.

Riordan array (1/sqrt(1-4x),x/sqrt(1-4x)). [From Paul Barry (pbarry(AT)wit.ie), May 06 2009]

a(n, 2*k+1) = binomial(n-k-1, k)*4^(n-2*k-1), a(n, 2*k) = binomial(2*(n-k), n-k)*binomial(n-k, k)/binomial(2*k, k), k >= 0, n >= m >= 0; a(n, m) := 0 if n<m.

Column recursion: a(n, m)=2*(2*n-m-1)*a(n-1, m)/(n-m), n>m >= 0, a(m, m) := 1.

G.f. for column m: cbie(x)*((x*cbie(x))^m, with cbie(x) := 1/sqrt(1-4*x).

G.f.: 1/(1-xy-2x/(1-x/(1-x/(1-x/(1-x/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), May 06 2009]

{1}; {2,1}; {6,4,1}; {20,16,6,1};...

Fourth row polynomial (n=3): p(3,x)= 20+16*x+6*x^2+x^3

Contribution from Paul Barry (pbarry(AT)wit.ie), May 06 2009: (Start)

Production matrix begins

2, 1,

2, 2, 1,

0, 2, 2, 1,

-2, 0, 2, 2, 1,

0, -2, 0, 2, 2, 1,

4, 0, -2, 0, 2, 2, 1,

0, 4, 0, -2, 0, 2, 2, 1,

-10, 0, 4, 0, -2, 0, 2, 2, 1,

0, -10, 0, 4, 0, -2, 0, 2, 2, 1 (End)

Cf. A000984, A035324, A054336 . Row sums: A026671(n).

Sequence in context: A073387 A125693 A094527 this_sequence A110681 A117852 A080245

Adjacent sequences: A054332 A054333 A054334 this_sequence A054336 A054337 A054338

easy,nice,nonn,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Mar 13 2000