A110292 Riordan array (1-u, u) where u=(-1 + sqrt(1+8*x))/4.
1, -1, 1, 2, -3, 1, -8, 12, -5, 1, 40, -60, 26, -7, 1, -224, 336, -148, 44, -9, 1, 1344, -2016, 896, -280, 66, -11, 1, -8448, 12672, -5664, 1824, -464, 92, -13, 1, 54912, -82368, 36960, -12144, 3240, -708, 122, -15, 1, -366080, 549120, -247104, 82368, -22704, 5280, -1020, 156, -17, 1
Offset: 0
Examples
Triangle begins as:
1;
-1, 1;
2, -3, 1;
-8, 12, -5, 1;
40, -60, 26, -7, 1;
-224, 336, -148, 44, -9, 1;
1344, -2016, 896, -280, 66, -11, 1;
-8448, 12672, -5664, 1824, -464, 92, -13, 1;
54912, -82368, 36960, -12144, 3240, -708, 122, -15, 1;
Links
- G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Programs
-
Magma
R
:=PowerSeriesRing(Rationals(), 30); F:= func< k | Coefficients(R!( (5-Sqrt(1+8*x))*(-1+Sqrt(1+8*x) )^k/4^(k+1) )) >; A110292:= func< n,k | F(k)[n-k+1] >; [A110292(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 04 2023 -
Mathematica
F[k_]:= CoefficientList[Series[(5-Sqrt[1+8*x])*(-1+Sqrt[1+8*x])^k/4^(k +1), {x,0,20}], x]; A110292[n_, k_]:= F[k][[n+1]]; Table[A110292[n, k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jan 04 2023 *) -
SageMath
def p(k,x): return (5-sqrt(1+8*x))*(-1+sqrt(1+8*x))^k/4^(k+1) def A110292(n,k): return ( p(k,x) ).series(x, 30).list()[n] flatten([[A110292(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Jan 04 2023
Formula
T(n, 0) = (-1)^n * 2^(n-1) * Catalan(n-1) + (3/2)*[n=0].
From G. C. Greubel, Jan 04 2023: (Start)
T(n, n) = 1.
T(n, n-1) = 1-2*n.
T(n, n-2) = 2*A028872(n).
T(n, 1) = (-1)^(n-1) * A181282(n-1), n >= 1.
Sum_{k=0..n} T(n, k) = A000007(n). (End)
Comments