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A208752 Triangle of coefficients of polynomials v(n,x) jointly generated with A208751; see the Formula section.

%I A208752 #22 Aug 03 2025 03:21:18
%S A208752 1,2,1,3,5,1,4,14,8,1,5,30,34,11,1,6,55,104,63,14,1,7,91,259,253,101,
%T A208752 17,1,8,140,560,806,504,148,20,1,9,204,1092,2178,1966,884,204,23,1,10,
%U A208752 285,1968,5202,6412,4090,1420,269,26,1,11,385,3333,11286,18238
%N A208752 Triangle of coefficients of polynomials v(n,x) jointly generated with A208751; see the Formula section.
%C A208752 For a discussion and guide to related arrays, see A208510.
%C A208752 Subtriangle of the triangle given by (0, 2, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 17 2012
%C A208752 Setting v(0,x) = 0, the sequence of polynomials {v(n,x) : n >= 0} satisfies the second-order recurrence v(n,x) = (x + 2)*v(n-1,x) + (x - 1)*v(n-2,x) with v(0,x) = 0 and v(1,x) = 1. Then by Norfleet, this sequence of polynomials is a strong divisibility sequence of polynomials in the ring Z[x], that is gcd(v(n,x), v(m,x)) = v(gcd(n,m),x). In particular, if n divides m then v(n,x) divides v(m,x) in Z[x]. - _Peter Bala_, Feb 07 2024
%H A208752 M. Norfleet, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/43-2.html">Characterization of second-order strong divisibility sequences of polynomials</a>, The Fibonacci Quarterly, 43(2) (2005), 166-169.
%F A208752 u(n,x)=u(n-1,x)+2x*v(n-1,x),
%F A208752 v(n,x)=u(n-1,x)+(x+1)*v(n-1,x),
%F A208752 where u(1,x)=1, v(1,x)=1.
%F A208752 T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-1), T(1,0) = 1, T(2,0) = 2, T(2,1) = 1, T(n,k) = 0 if k<0 or if k>=n. - _Philippe Deléham_, Mar 17 2012
%F A208752 G.f.: -x*y/(-1+2*x-x^2+x^2*y+x*y). - _R. J. Mathar_, Aug 12 2015
%e A208752 First five rows:
%e A208752   1
%e A208752   2   1
%e A208752   3   5    1
%e A208752   4   14   8    1
%e A208752   5   30   34   11   1
%e A208752 First five polynomials u(n,x) - see A208751:
%e A208752   1
%e A208752   1 + 2*x
%e A208752   1 + 6*x + 2*x^2
%e A208752   1 + 12*x + 12*x^2 + 2*x^3
%e A208752   1 + 20*x + 40*x^2 + 18*x^3 + 2*x^4
%e A208752 (0, 2, -1/2, 1/2, 0, 0, ...) DELTA (1, 0, 1/2, -1/2, 0, 0, ...) begins:
%e A208752   1
%e A208752   0, 1
%e A208752   0, 2, 1
%e A208752   0, 3, 5, 1
%e A208752   0, 4, 14, 8, 1
%e A208752   0, 5, 30, 34, 11, 1. - _Philippe Deléham_, Mar 17 2012
%t A208752 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208752 u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
%t A208752 v[n_, x_] := u[n - 1, x] + (x + 1) v[n - 1, x];
%t A208752 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208752 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208752 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208752 TableForm[cu]
%t A208752 Flatten[%]    (* A208751 *)
%t A208752 Table[Expand[v[n, x]], {n, 1, z}]
%t A208752 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208752 TableForm[cv]
%t A208752 Flatten[%]    (* A208752 *)
%Y A208752 Cf. A208751, A208510.
%K A208752 nonn,tabl
%O A208752 1,2
%A A208752 _Clark Kimberling_, Mar 01 2012