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A210552 Triangle of coefficients of polynomials u(n,x) jointly generated with A210553; see the Formula section. 3

%I

%S 1,1,2,1,3,3,1,4,5,5,1,5,7,10,8,1,6,9,16,18,13,1,7,11,23,31,33,21,1,8,

%T 13,31,47,62,59,34,1,9,15,40,66,101,119,105,55,1,10,17,50,88,151,205,

%U 227,185,89,1,11,19,61,113,213,321,414,426,324,144,1,12,21,73

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A210553; see the Formula section.

%C Let T(n,k) denote the term in row n, column k.

%C T(n,n): A000045 (Fibonacci numbers)

%C T(n,n-1): A010049 (second-order Fibonacci numbers)

%C T(n,1): 1,1,1,1,1,1,1,1,1,1,1,,...

%C T(n,2): 2,3,4,5,6,7,8,9,10,11,...

%C T(n,3): 3,5,7,9,11,13,15,17,19,...

%C T(n,4): A052905

%C Row sums: A000225

%C Alternating row sums: A094024 (signed)

%C For a discussion and guide to related arrays, see A208510.

%F u(n,x)=x*u(n-1,x)+x*v(n-1,x)+1,

%F v(n,x)=x*u(n-1,x)+v(n-1,x)+1,

%F where u(1,x)=1, v(1,x)=1.

%e First five rows:

%e 1

%e 1...2

%e 1...3...3

%e 1...4...5...5

%e 1...5...7...10...8

%e First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 3x^2.

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

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

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

%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[%] (* A210552 *)

%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[%] (* A210553 *)

%t Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)

%t Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)

%t Table[u[n, x] /. x -> -1, {n, 1, z}] (* A094024 *)

%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* A052551 *)

%Y Cf. A210553, A208510.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 22 2012

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Last modified November 18 14:59 EST 2019. Contains 329262 sequences. (Running on oeis4.)