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 Pell(z)/(1-x*z*Pell(z)) with Pell(x)=1/(1-2*x-x^2) = g.f. for A000129(n+1) (Pell numbers without 0).

Column sequences are A000129(n+1), A006645(n+1), A054457(n) for m=0..2.

Riordan array (1/(1-2x-x^2),x/(1-2x-x^2)). - Paul Barry (pbarry(AT)wit.ie), Mar 15 2005

As a Riordan array, this factors as (1/(1-x^2),x/(1-x^2))*(1/(1-2x),x/(1-2x)), [abs(A049310) times square of A007318, or A038207]. - Paul Barry (pbarry(AT)wit.ie), Jul 28 2005

Coefficients of polynomials defined by P(x, 0) = 1; P(x, 1) = 2 - x; P(x, n) = (2 - x)*P(x, n - 1) + P(x, n - 2). - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 24 2008

a(n, m) := ((n-m+1)*a(n, m-1) + (n+m)*a(n-1, m-1))/(4*m), n >= m >= 1, a(n, 0)= P(n+1)= A000129(n+1) (Pell numbers without P(0)), a(n, m) := 0 if n<m.

G.f. for column m: Pell(x)*(x*Pell(x))^m, m >= 0, with Pell(x) G.f. for A000129(n+1).

Number triangle T(n, k) with T(n, 0)=A000129(n), T(1, 1)=1, T(n, k)=0 if k>n, T(n, k)=T(n-1, k-1)+T(n-2, k)+2T(n-1, k) otherwise; T(n, k)=if(k<=n, sum{j=0..floor((n-k)/2), C(n-j, k)C(n-k-j, j)2^(n-2j-k)}; - Paul Barry (pbarry(AT)wit.ie), Mar 15 2005

{1}; {2,1}; {5,4,1}; {12,14,6,1};...

Fourth row polynomial (n=3): p(3,x)= 12+14*x+6*x^2+x^3

Triangle begins:

{1},

{2, -1},

{5, -4, 1},

{12, -14, 6, -1},

{29, -44, 27, -8, 1},

{70, -131,104, -44, 10, -1},

{169, -376, 366, -200, 65, -12, 1},

{408, -1052, 1212, -810, 340, -90, 14, -1},

{985, -2888, 3842, -3032, 1555, -532, 119, -16, 1},

{2378, -7813, 11784, -10716, 6482, -2709, 784, -152, 18, -1},

{5741, -20892, 35223, -36248, 25235, -12432, 4396, -1104, 189, -20, 1},

{13860, -55338, 103122, -118435, 93200, -52808, 22008, -6756, 1500, -230, 22, -1},

{33461, -145428, 296805, -376240, 330070, -211248, 101220, -36624, 9945, -1980,275, -24, 1}

P[x, 0] = 1; P[x, 1] = 2 - x; P[x_, n_] := P[x, n] = (2 - x)*P[x, n - 1] + P[x, n - 2] Table[ExpandAll[P[x, n]], {n, 0, 12}]; a2 = Table[CoefficientList[P[x, n], x], {n, 0, 12}]; Flatten[a2] Table[Apply[Plus, CoefficientList[P[x, n], x]], {n, 0, 12}]; - Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 24 2008

Cf. A000129. Row sums: A006190(n+1).

Cf. A129844.

Sequence in context: A110552 A129161 A103415 this_sequence A096164 A104710 A039598

Adjacent sequences: A054453 A054454 A054455 this_sequence A054457 A054458 A054459

easy,nonn,tabl

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 27 2000 and May 08 2000.