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A207536 Triangle of coefficients of polynomials u(n,x) jointly generated with A105070; see Formula section.

%I A207536 #16 Jan 22 2020 17:37:24
%S A207536 1,1,2,1,6,1,12,4,1,20,20,1,30,60,8,1,42,140,56,1,56,280,224,16,1,72,
%T A207536 504,672,144,1,90,840,1680,720,32,1,110,1320,3696,2640,352,1,132,1980,
%U A207536 7392,7920,2112,64,1,156,2860,13728,20592,9152,832,1,182,4004
%N A207536 Triangle of coefficients of polynomials u(n,x) jointly generated with A105070; see Formula section.
%C A207536 Subtriangle of the triangle given by (1, 0, 1, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Apr 08 2012
%F A207536 u(n,x) = u(n-1,x) + 2x*v(n-1,x) and v(n,x) = u(n-1,x) + v(n-1,x),
%F A207536 where u(1,x)=1, v(1,x)=1.
%F A207536 From _Philippe Deléham_, Apr 08 2012: (Start)
%F A207536 As DELTA-triangle T(n,k) with 0 <= k <= n:
%F A207536 G.f.: (1-x)/(1-2*x+x^2-2*y*x^2).
%F A207536 T(n,k) = 2*T(n-1,k) - T(n-2,k) + 2*T(n-2,k-1), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k < 0 or if k > n.
%F A207536 T(n,k) = A034839(n,k)*2^k = binomial(n,2*k)*2^k . (End)
%e A207536 First seven rows:
%e A207536   1;
%e A207536   1,  2;
%e A207536   1,  6,
%e A207536   1, 12,   4;
%e A207536   1, 20,  20,
%e A207536   1, 30,  60,  8;
%e A207536   1, 42, 140, 56;
%e A207536 From _Philippe Deléham_, Apr 08 2012: (Start)
%e A207536 (1, 0, 1, 0, 0, 0, 0, ...) DELTA (0, 2, -2, 0, 0, 0, 0, ...) begins:
%e A207536   1;
%e A207536   1,  0;
%e A207536   1,  2,   0;
%e A207536   1,  6,   0,  0;
%e A207536   1, 12,   4,  0, 0;
%e A207536   1, 20,  20,  0, 0, 0;
%e A207536   1, 30,  60,  8, 0, 0, 0;
%e A207536   1, 42, 140, 56, 0, 0, 0, 0; (End)
%t A207536 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A207536 u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x]
%t A207536 v[n_, x_] := u[n - 1, x] + v[n - 1, x]
%t A207536 Table[Factor[u[n, x]], {n, 1, z}]
%t A207536 Table[Factor[v[n, x]], {n, 1, z}]
%t A207536 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A207536 TableForm[cu]
%t A207536 Flatten[%]  (* A207536 *)
%t A207536 Table[Expand[v[n, x]], {n, 1, z}]
%t A207536 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A207536 TableForm[cv]
%t A207536 Flatten[%]  (* A105070 *)
%Y A207536 Cf. A105070.
%K A207536 nonn,tabf
%O A207536 1,3
%A A207536 _Clark Kimberling_, Feb 18 2012