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A110292 Riordan array (1-u, u) where u=(-1 + sqrt(1+8*x))/4.

%I A110292 #10 Jan 06 2023 09:06:15
%S A110292 1,-1,1,2,-3,1,-8,12,-5,1,40,-60,26,-7,1,-224,336,-148,44,-9,1,1344,
%T A110292 -2016,896,-280,66,-11,1,-8448,12672,-5664,1824,-464,92,-13,1,54912,
%U A110292 -82368,36960,-12144,3240,-708,122,-15,1,-366080,549120,-247104,82368,-22704,5280,-1020,156,-17,1
%N A110292 Riordan array (1-u, u) where u=(-1 + sqrt(1+8*x))/4.
%C A110292 Inverse of Riordan array (1/(1-x), x*(1+2*x)), A110291.
%H A110292 G. C. Greubel, <a href="/A110292/b110292.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A110292 T(n, 0) = (-1)^n * 2^(n-1) * Catalan(n-1) + (3/2)*[n=0].
%F A110292 From _G. C. Greubel_, Jan 04 2023: (Start)
%F A110292 T(n, n) = 1.
%F A110292 T(n, n-1) = 1-2*n.
%F A110292 T(n, n-2) = 2*A028872(n).
%F A110292 T(n, 1) = (-1)^(n-1) * A181282(n-1), n >= 1.
%F A110292 Sum_{k=0..n} T(n, k) = A000007(n). (End)
%e A110292 Triangle begins as:
%e A110292       1;
%e A110292      -1,      1;
%e A110292       2,     -3,     1;
%e A110292      -8,     12,    -5,      1;
%e A110292      40,    -60,    26,     -7,    1;
%e A110292    -224,    336,  -148,     44,   -9,    1;
%e A110292    1344,  -2016,   896,   -280,   66,  -11,   1;
%e A110292   -8448,  12672, -5664,   1824, -464,   92, -13,   1;
%e A110292   54912, -82368, 36960, -12144, 3240, -708, 122, -15,  1;
%t A110292 F[k_]:= CoefficientList[Series[(5-Sqrt[1+8*x])*(-1+Sqrt[1+8*x])^k/4^(k +1), {x,0,20}], x];
%t A110292 A110292[n_, k_]:= F[k][[n+1]];
%t A110292 Table[A110292[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jan 04 2023 *)
%o A110292 (Magma)
%o A110292 R<x>:=PowerSeriesRing(Rationals(), 30);
%o A110292 F:= func< k | Coefficients(R!( (5-Sqrt(1+8*x))*(-1+Sqrt(1+8*x) )^k/4^(k+1) )) >;
%o A110292 A110292:= func< n,k | F(k)[n-k+1] >;
%o A110292 [A110292(n,k): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Jan 04 2023
%o A110292 (SageMath)
%o A110292 def p(k,x): return (5-sqrt(1+8*x))*(-1+sqrt(1+8*x))^k/4^(k+1)
%o A110292 def A110292(n,k): return ( p(k,x) ).series(x, 30).list()[n]
%o A110292 flatten([[A110292(n,k) for k in range(n+1)] for n in range(13)]) # _G. C. Greubel_, Jan 04 2023
%Y A110292 Cf. A000007, A028872, A110291, A181282.
%K A110292 easy,sign,tabl
%O A110292 0,4
%A A110292 _Paul Barry_, Jul 18 2005