A054456 Convolution triangle of A000129(n) (Pell numbers).

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

Offset: 0

Author

Wolfdieter Lang, Apr 27 2000 and May 08 2000

Keywords

• easy
• nonn
• tabl

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 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, 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, 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, Mar 24 2008

Subtriangle (obtained by dropping the first column) of the triangle given by (0, 2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Feb 19 2013

T(n,k) is the number of words of length n over {0,1,2,3} having k letters 3 and avoiding runs of odd length of the letter 0. - Milan Janjic, Jan 14 2017

Examples

			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},
The triangle (0, 2, 1/2, -1/2, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, ...) begins:
1
0, 1
0, 2, 1
0, 5, 4, 1
0, 12, 14, 6, 1
0, 29, 44, 27, 8, 1 - _Philippe Deléham_, Feb 19 2013
		

Links

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

Crossrefs

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

Cf. A129844.

Programs

Maple

G := 1/(1-(x+2)*z-z^2): Gser := simplify(series(G, z = 0, 18)): for n from 0 to 15 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 15 do seq(coeff(P[n], x, j), j = 0 .. n) end do; # yields sequence in triangular form - Emeric Deutsch, Aug 30 2015
T := (n,k) -> `if`(n=0,1,2^(n-k)*binomial(n,k)*hypergeom([(k-n)/2,(k-n+1)/2],[-n], -1)): seq(seq(simplify(T(n,k)),k=0..n),n=0..10); # Peter Luschny, Apr 25 2016
# Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
PMatrix(10, A000129); # Peter Luschny, Oct 19 2022

Mathematica

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[Abs@ CoefficientList[P[x, n], x], {n, 0, 10}] // Flatten (* Roger L. Bagula, Mar 24 2008, edited by Michael De Vlieger, Apr 25 2018 *)

Formula

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

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, Mar 15 2005

Bivariate g.f. G(x,z) = 1/[1 - (2 + x)z - z^2]. G.f. for column k = z^k/(1 - 2z - z^2)^{k+1} (k>=0). - Emeric Deutsch, Aug 30 2015

T(n,k) = 2^(n-k)*C(n,k)*hypergeom([(k-n)/2,(k-n+1)/2],[-n],-1)) for n>=1. - Peter Luschny, Apr 25 2016

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