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A208525 Triangle of coefficients of polynomials v(n,x) jointly generated with A208524; see the Formula section.

%I A208525 #5 Mar 30 2012 18:58:13
%S A208525 1,2,3,3,7,5,4,12,18,11,5,18,42,49,21,6,25,80,135,116,43,7,33,135,295,
%T A208525 381,279,85,8,42,210,560,966,1050,638,171,9,52,308,966,2086,2996,2724,
%U A208525 1453,341,10,63,432,1554,4032,7182,8688,6921,3240,683,11,75
%N A208525 Triangle of coefficients of polynomials v(n,x) jointly generated with A208524; see the Formula section.
%C A208525 Alternating row sums: 1,-1,1,-1,1,-1,1,-1,...
%F A208525 u(n,x)=u(n-1,x)+x*v(n-1,x),
%F A208525 v(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x)+1,
%F A208525 where u(1,x)=1, v(1,x)=1.
%e A208525 First five rows:
%e A208525 1
%e A208525 2...3
%e A208525 3...7....5
%e A208525 4...12...18...11
%e A208525 5...18...42...49...21
%e A208525 First five polynomials v(n,x):
%e A208525 1
%e A208525 2 + 3x
%e A208525 3 + 7x + 5x^2
%e A208525 4 + 12x + 18x^2 + 11x^3
%e A208525 5 + 18x + 42x^2 + 49x^3 + 21x^4
%t A208525 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208525 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208525 v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t A208525 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208525 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208525 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208525 TableForm[cu]
%t A208525 Flatten[%]  (* A208524 *)
%t A208525 Table[Expand[v[n, x]], {n, 1, z}]
%t A208525 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208525 TableForm[cv]
%t A208525 Flatten[%]  (* A208525 *)
%t A208525 Table[u[n, x] /. x -> 1, {n, 1, z}] (*A060816*)
%t A208525 Table[v[n, x] /. x -> 1, {n, 1, z}] (*|A084244|*)
%t A208525 Table[u[n, x] /. x -> -1, {n, 1, z}] (*alt. row sums*)
%t A208525 Table[v[n, x] /. x -> -1, {n, 1, z}] (*alt. row sums*)
%Y A208525 Cf. A208524.
%K A208525 nonn,tabl
%O A208525 1,2
%A A208525 _Clark Kimberling_, Feb 29 2012