cp's OEIS Frontend

A207636 Triangle of coefficients of polynomials v(n,x) jointly generated with A207635; see Formula section.

%I A207636 #14 Jan 22 2020 20:12:30
%S A207636 1,3,2,6,7,2,12,20,11,2,24,52,42,15,2,48,128,136,72,19,2,96,304,400,
%T A207636 280,110,23,2,192,704,1104,960,500,156,27,2,384,1600,2912,3024,1960,
%U A207636 812,210,31,2,768,3584,7424,8960,6944,3584,1232,272,35,2,1536,7936
%N A207636 Triangle of coefficients of polynomials v(n,x) jointly generated with A207635; see Formula section.
%C A207636 As triangle T(n,k) with 0 <= k <= n, it is (3, -1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Feb 26 2012
%F A207636 u(n,x) = u(n-1,x) + v(n-1,x),
%F A207636 v(n,x) = (x+1)*u(n-1,x) + (x+1)*v(n-1,x) + 1,
%F A207636 where u(1,x)=1, v(1,x)=1.
%F A207636 From _Philippe Deléham_, Feb 26 2012: (Start)
%F A207636 As triangle T(n,k), 0 <= k <= n:
%F A207636 T(n,k) = 2*T(n-1,k) + T(n-1,k-1) with T(0,0) = 1, T(1,0) = 3, T(1,1) = 2 and T(n,k) = 0 if k < 0 or if k > n.
%F A207636 G.f.: (1+x+y*x)/(1-2*x-y*x).
%F A207636 Sum_{k=0..n} T(n,k)*x^k = A003945(n), |A084244(n)|, A189274(n) for x = 0, 1, 3 respectively.
%F A207636 Sum_{k=0..n} T(n,k)*x^(n-k) = A040000(n), |A084244(n)|, A128625(n) for x = 0, 1, 2 respectively. (End)
%e A207636 First five rows:
%e A207636    1;
%e A207636    3,  2;
%e A207636    6,  7,  2;
%e A207636   12, 20, 11,  2;
%e A207636   24, 52, 42, 15,  2;
%t A207636 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A207636 u[n_, x_] := u[n - 1, x] + v[n - 1, x]
%t A207636 v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1
%t A207636 Table[Factor[u[n, x]], {n, 1, z}]
%t A207636 Table[Factor[v[n, x]], {n, 1, z}]
%t A207636 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A207636 TableForm[cu]
%t A207636 Flatten[%]  (* A207635 *)
%t A207636 Table[Expand[v[n, x]], {n, 1, z}]
%t A207636 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A207636 TableForm[cv]
%t A207636 Flatten[%]  (* A207636 *)
%Y A207636 Cf. A207635.
%Y A207636 Cf. A084938, A003945, A040000.
%K A207636 nonn,tabl
%O A207636 1,2
%A A207636 _Clark Kimberling_, Feb 24 2012