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A207537 Triangle of coefficients of polynomials u(n,x) jointly generated with A207538; see Formula section.

%I A207537 #24 Feb 03 2021 17:45:43
%S A207537 1,2,1,4,3,8,8,1,16,20,5,32,48,18,1,64,112,56,7,128,256,160,32,1,256,
%T A207537 576,432,120,9,512,1280,1120,400,50,1,1024,2816,2816,1232,220,11,2048,
%U A207537 6144,6912,3584,840,72,1,4096,13312,16640,9984,2912,364,13
%N A207537 Triangle of coefficients of polynomials u(n,x) jointly generated with A207538; see Formula section.
%C A207537 Another version in A201701. - _Philippe Deléham_, Mar 03 2012
%C A207537 Subtriangle of the triangle given by (1, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 03 2012
%C A207537 Columns: A011782, A001792, A001793, A001794, A006974, A006975, A006976. - _Philippe Deléham_, Mar 03 2012
%C A207537 Diagonal sums: A052980. - _Philippe Deléham_, Mar 03 2012
%F A207537 u(n,x) = u(n-1,x) + (x+1)*v(n-1,x), v(n,x) = u(n-1,x) + v(n-1,x), where u(1,x)=1, v(1,x)=1.  Also, A207537 = |A028297|.
%F A207537 T(n,k) = 2*T(n-1,k) + T(n-2,k-1). - _Philippe Deléham_, Mar 03 2012
%F A207537 G.f.: -(1+x*y)*x*y/(-1+2*x+x^2*y). - _R. J. Mathar_, Aug 11 2015
%F A207537 T(n, k) = [x^k] hypergeom([-n/2, -n/2 + 1/2], [1/2], x + 1) provided offset is set to 0 and 1 prepended. - _Peter Luschny_, Feb 03 2021
%e A207537 First seven rows:
%e A207537    1;
%e A207537    2,   1;
%e A207537    4,   3;
%e A207537    8,   8,  1;
%e A207537   16,  20,  5,
%e A207537   32,  48, 18, 1;
%e A207537   64, 112, 56, 7;
%e A207537 From _Philippe Deléham_, Mar 03 2012: (Start)
%e A207537 Triangle A201701 begins:
%e A207537    1;
%e A207537    1,   0;
%e A207537    2,   1,  0;
%e A207537    4,   3,  0, 0;
%e A207537    8,   8,  1, 0, 0;
%e A207537   16,  20,  5, 0, 0, 0;
%e A207537   32,  48, 18, 1, 0, 0, 0;
%e A207537   64, 112, 56, 7, 0, 0, 0, 0;
%e A207537   ... (End)
%t A207537 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A207537 u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x]
%t A207537 v[n_, x_] := u[n - 1, x] + v[n - 1, x]
%t A207537 Table[Factor[u[n, x]], {n, 1, z}]
%t A207537 Table[Factor[v[n, x]], {n, 1, z}]
%t A207537 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A207537 TableForm[cu]
%t A207537 Flatten[%]  (* A207537, |A028297| *)
%t A207537 Table[Expand[v[n, x]], {n, 1, z}]
%t A207537 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A207537 TableForm[cv]
%t A207537 Flatten[%]  (* A207538, |A133156| *)
%t A207537 (* Prepending 1 and with offset 0: *)
%t A207537 Tpoly[n_] := HypergeometricPFQ[{-n/2, -n/2 + 1/2}, {1/2}, x + 1];
%t A207537 Table[CoefficientList[Tpoly[n], x], {n, 0, 12}] // Flatten (* _Peter Luschny_, Feb 03 2021 *)
%Y A207537 Cf. A028297, A207538, A133156.
%K A207537 nonn,tabf
%O A207537 1,2
%A A207537 _Clark Kimberling_, Feb 18 2012