cp's OEIS Frontend

A208750 Triangle of coefficients of polynomials v(n,x) jointly generated with A208749; see the Formula section.

%I A208750 #14 Jan 24 2020 03:30:22
%S A208750 1,2,1,3,4,2,4,11,10,2,5,24,32,16,4,6,45,84,72,32,4,7,76,194,240,156,
%T A208750 48,8,8,119,406,666,592,300,88,8,9,176,784,1632,1896,1344,576,128,16,
%U A208750 10,249,1416,3648,5344,4904,2848,1024,224,16,11,340,2418,7584
%N A208750 Triangle of coefficients of polynomials v(n,x) jointly generated with A208749; see the Formula section.
%C A208750 For a discussion and guide to related arrays, see A208510.
%C A208750 Subtriangle of the triangle T(n,k) given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 16 2012
%F A208750 u(n,x) = u(n-1,x) + 2x*v(n-1,x),
%F A208750 v(n,x) = (x+1)*u(n-1,x) + v(n-1,x),
%F A208750 where u(1,x)=1, v(1,x)=1.
%F A208750 As DELTA-triangle with 0 <= k <= n: g.f.: (1-x+x^2-y*x^2-2*t^2*x^2)/(1-2*x+x^2-2*y*x^2-2*y^2*x^2). - _Philippe Deléham_, Mar 16 2012
%F A208750 As DELTA-triangle: T(n,k) = 2*T(n-1,k) - T(n-2,k) + 2*T(n-2,k-1) + 2*T(n-2,k-2), T(0,0) = T(1,0) = T(2,1) = 1, T(2,0) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. - _Philippe Deléham_, Mar 16 2012
%e A208750 First five rows:
%e A208750   1;
%e A208750   2,  1;
%e A208750   3,  4,  2;
%e A208750   4, 11, 10,  2;
%e A208750   5, 24, 32, 16,  4;
%e A208750 First five polynomials v(n,x):
%e A208750   1
%e A208750   2 +   x
%e A208750   3 +  4x +  2x^2
%e A208750   4 + 11x + 10x^2 +  2x^3
%e A208750   5 + 24x + 32x^2 + 16x^3 + 4x^4
%e A208750 From _Philippe Deléham_, Mar 16 2012: (Start)
%e A208750 (1, 1, -1, 1, 0, 0, ...) DELTA (0, 1, 1, -2, 0, 0, ...) begins:
%e A208750   1;
%e A208750   1,  0;
%e A208750   2,  1,  0;
%e A208750   3,  4,  2,  0;
%e A208750   4, 11, 10,  2,  0;
%e A208750   5, 24, 32, 16,  4,  0;
%e A208750   6, 45, 84, 72, 32,  4,  0; (End)
%t A208750 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208750 u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
%t A208750 v[n_, x_] := (x + 1)*u[n - 1, x] + v[n - 1, x];
%t A208750 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208750 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208750 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208750 TableForm[cu]
%t A208750 Flatten[%]    (* A208749 *)
%t A208750 Table[Expand[v[n, x]], {n, 1, z}]
%t A208750 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208750 TableForm[cv]
%t A208750 Flatten[%]    (* A208750 *)
%Y A208750 Cf. A208749, A208510.
%K A208750 nonn,tabl
%O A208750 1,2
%A A208750 _Clark Kimberling_, Mar 02 2012