A054458 Convolution triangle based on A001333(n), n >= 1.

1, 3, 1, 7, 6, 1, 17, 23, 9, 1, 41, 76, 48, 12, 1, 99, 233, 204, 82, 15, 1, 239, 682, 765, 428, 125, 18, 1, 577, 1935, 2649, 1907, 775, 177, 21, 1, 1393, 5368, 8680, 7656, 4010, 1272, 238, 24, 1, 3363, 14641, 27312, 28548, 18358, 7506, 1946, 308, 27, 1

Offset

0,2

Comments

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 LPell(z)/(1-x*z*LPell(z)) with LPell(z) given in 'Formula'.

Column sequences are A001333(n+1), A054459(n), A054460(n) for m=0..2.

Mirror image of triangle in A209696. - Philippe Deléham, Mar 24 2012

Subtriangle of the triangle given by (0, 3, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 25 2012

Riordan array ((1+x)/(1-2*x-x^2), (x+x^2)/(1-2*x-x^2)). - Philippe Deléham, Mar 25 2012

Links

Table of n, a(n) for n=0..54.

Milan Janjić, Words and Linear Recurrences, J. Int. Seq. 21 (2018), #18.1.4.

Formula

a(n, m) := ((n-m+1)*a(n, m-1) + (2n-m)*a(n-1, m-1) + (n-1)*a(n-2, m-1))/(4*m), n >= m >= 1; a(n, 0)= A001333(n+1); a(n, m) := 0 if n<m.

G.f. for column m: LPell(x)*(x*LPell(x))^m, m >= 0, with LPell(x)= (1+x)/(1-2*x-x^2) = g.f. for A001333(n+1).

G.f.: (1+x)/(1-2*x-y*x-x^2-y*x^2). - Philippe Deléham, Mar 25 2012

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(0,0) = T(1,1) = 1, T(1,0) = 3 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 25 2012

Sum_{k=0..n} T(n,k)*x^k = A040000(n), A001333(n+1), A055099(n), A126473(n), A126501(n), A126528(n) for x = -1, 0, 1, 2, 3, 4 respectively. - Philippe Deléham, Mar 25 2012

Example

Fourth row polynomial (n=3): p(3,x)= 17+23*x+9*x^2+x^3.
Triangle begins :
  1
  3, 1
  7, 6, 1
  17, 23, 9, 1
  41, 76, 48, 12, 1
  99, 233, 204, 82, 15, 1
  239, 682, 765, 428, 125, 18, 1. - Philippe Deléham, Mar 25 2012
(0, 3, -2/3, -1/3, 0, 0, 0, ...) DELTA (1, 0, 0, 0, ...) begins :
  1
  0, 1
  0, 3, 1
  0, 7, 6, 1
  0, 17, 23, 9, 1
  0, 41, 76, 48, 12, 1
  0, 99, 233, 204, 82, 15, 1
  0, 239, 682, 765, 428, 125, 15, 1. - Philippe Deléham, Mar 25 2012

Crossrefs

Cf. A002203(n+1)/2. Row sums: A055099(n).

Cf. A001333, A054459, A054460.

Cf. A040000, A001333, A055099, A126473, A126501, A126528.

Sequence in context: A111806 A372118 A321163 * A110168 A323663 A355928

Adjacent sequences: A054455 A054456 A054457 * A054459 A054460 A054461

Keyword

easy,nonn,tabl

Author

Wolfdieter Lang, Apr 26 2000

Status

approved