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A054456 Convolution triangle of A000129(n) (Pell numbers). 17
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 (list; table; graph; refs; listen; history; text; internal format)
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 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
LINKS
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) + (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, 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
EXAMPLE
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
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 *)
CROSSREFS
Cf. A000129. Row sums: A006190(n+1).
Cf. A129844.
Sequence in context: A110552 A129161 A103415 * A096164 A201166 A318942
KEYWORD
easy,nonn,tabl
AUTHOR
Wolfdieter Lang, Apr 27 2000 and May 08 2000
STATUS
approved

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Last modified March 29 09:28 EDT 2024. Contains 371268 sequences. (Running on oeis4.)