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

0,4

Comments

Inverse of Riordan array (1/(1-x), x*(1+2*x)), A110291.

Links

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

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)

Example

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;

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 *)

Prog

(Magma)
R<x>:=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
(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

Crossrefs

Cf. A000007, A028872, A110291, A181282.

Sequence in context: A121634 A006015 A301332 * A138672 A103749 A098435

Adjacent sequences: A110289 A110290 A110291 * A110293 A110294 A110295

Keyword

easy,sign,tabl

Author

Paul Barry, Jul 18 2005

Status

approved