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%I #51 Oct 19 2022 11:07:54
%S 1,2,1,5,4,1,12,14,6,1,29,44,27,8,1,70,131,104,44,10,1,169,376,366,
%T 200,65,12,1,408,1052,1212,810,340,90,14,1,985,2888,3842,3032,1555,
%U 532,119,16,1,2378,7813,11784,10716,6482,2709,784,152,18,1,5741,20892,35223
%N Convolution triangle of A000129(n) (Pell numbers).
%C 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.
%C 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).
%C Column sequences are A000129(n+1), A006645(n+1), A054457(n) for m=0..2.
%C Riordan array (1/(1-2x-x^2),x/(1-2x-x^2)). - _Paul Barry_, Mar 15 2005
%C 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
%C 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
%C 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
%C 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
%H Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Janjic/janjic93.html">Words and Linear Recurrences</a>, J. Int. Seq. 21 (2018), #18.1.4.
%F 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.
%F G.f. for column m: Pell(x)*(x*Pell(x))^m, m >= 0, with Pell(x) G.f. for A000129(n+1).
%F 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
%F 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
%F 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
%e Fourth row polynomial (n=3): p(3,x)= 12+14*x+6*x^2+x^3
%e Triangle begins:
%e {1},
%e {2, 1},
%e {5, 4, 1},
%e {12, 14, 6, 1},
%e {29, 44, 27, 8, 1},
%e {70, 131,104, 44, 10, 1},
%e {169, 376, 366, 200, 65, 12, 1},
%e {408, 1052, 1212, 810, 340, 90, 14, 1},
%e {985, 2888, 3842, 3032, 1555, 532, 119, 16, 1},
%e {2378, 7813, 11784, 10716, 6482, 2709, 784, 152, 18, 1},
%e {5741, 20892, 35223, 36248, 25235, 12432, 4396, 1104, 189, 20, 1},
%e The triangle (0, 2, 1/2, -1/2, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, ...) begins:
%e 1
%e 0, 1
%e 0, 2, 1
%e 0, 5, 4, 1
%e 0, 12, 14, 6, 1
%e 0, 29, 44, 27, 8, 1 - _Philippe Deléham_, Feb 19 2013
%p 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
%p 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
%p # Uses function PMatrix from A357368. Adds column 1,0,0,0,... to the left.
%p PMatrix(10, A000129); # _Peter Luschny_, Oct 19 2022
%t 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 *)
%Y Cf. A000129. Row sums: A006190(n+1).
%Y Cf. A129844.
%K easy,nonn,tabl
%O 0,2
%A _Wolfdieter Lang_, Apr 27 2000 and May 08 2000
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