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Coefficients of a generalized Pell-Lucas polynomial read by rows.
8

%I #42 Dec 03 2021 19:25:34

%S 1,2,1,3,3,1,5,7,4,1,8,15,12,5,1,13,30,31,18,6,1,21,58,73,54,25,7,1,

%T 34,109,162,145,85,33,8,1,55,201,344,361,255,125,42,9,1,89,365,707,

%U 850,701,413,175,52,10,1,144,655,1416,1918,1806,1239,630,236,63,11,1

%N Coefficients of a generalized Pell-Lucas polynomial read by rows.

%C A122075 is jointly generated with A037027 as an array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=u(n-1,x)+(x+1)*v(n-1)x and v(n,x)=u(n-1,x)+x*v(n-1,x). See the Mathematica section. - _Clark Kimberling_, Mar 05 2012

%C Subtriangle of the triangle T(n,k) given by (1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 11 2012

%H G. C. Greubel, <a href="/A122075/b122075.txt">Table of n, a(n) for the first 50 rows, flattened</a>

%H Tian-Xiao He and Peter J.-S. Shiue, <a href="https://doi.org/10.3934/era.2021049">Identities for linear recursive sequences of order 2</a>, Elect. Res. Archive (2021) Vol. 29, No. 5, 3489-3507.

%H Tian-Xiao He, Peter J.-S. Shiue, Zihan Nie, and Minghao Chen, <a href="https://doi.org/10.3934/era.2020057">Recursive sequences and Girard-Waring identities with applications in sequence transformation</a>, Electronic Research Archive (2020) Vol. 28, No. 2, 1049-1062.

%H Y. Sun, <a href="https://www.fq.math.ca/Papers1/43-4/paper43-4-10b.pdf">Numerical Triangles and Several Classical Sequences</a>, Fib. Quart. 43, no. 4, (2005) 359-370.

%F T(n,k)=sum_(j=0..n-k+1) binomial(n-k-j+1,j)*binomial(n-j,k).

%F sum_(k>=0) T(n-k,k)=2^n.

%F sum_(k>=0) (-1)^k T(n-k,k)=2-delta(0,n).

%F G.f.: -(1+x)/(-1+x*y+x+x^2). - _R. J. Mathar_, Aug 11 2015

%e Triangle begins

%e 1

%e 2 1

%e 3 3 1

%e 5 7 4 1

%e 8 15 12 5 1

%e 13 30 31 18 6 1

%e A055830 = (1, 1, -1, 0, 0, 0, ...) DELTA ((0, 1, 0, 0, 0, 0, ...) begins :

%e 1

%e 1, 0

%e 2, 1, 0

%e 3, 3, 1, 0

%e 5, 7, 4, 1, 0

%e 8, 15, 12, 5, 1, 0

%e 13, 30, 31, 18, 6, 1, 0

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

%t u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];

%t v[n_, x_] := u[n - 1, x] + x*v[n - 1, x];

%t Table[Expand[u[n, x]], {n, 1, z/2}]

%t Table[Expand[v[n, x]], {n, 1, z/2}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%] (* A122075 *)

%t Table[Expand[v[n, x]], {n, 1, z}]

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%] (* A037027 *)

%t (* _Clark Kimberling_, Mar 05 2012 *)

%t CoefficientList[CoefficientList[Series[-(1 + x)/(-1 + x*y + x + x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* _G. C. Greubel_, Dec 24 2017 *)

%o (PARI) T(n,k)={ sum(j=0,n-k+1, binomial(n-k-j+1,j)*binomial(n-j,k)) ; } { nmax=10 ; for(n=0,nmax, for(k=0,n, print1(T(n,k),",") ; ); ); }

%Y See A055830 for another version.

%K easy,nonn,tabl

%O 0,2

%A _R. J. Mathar_, Oct 16 2006