A127895 Riordan array (1/(1+x)^3, x/(1+x)^3).

1, -3, 1, 6, -6, 1, -10, 21, -9, 1, 15, -56, 45, -12, 1, -21, 126, -165, 78, -15, 1, 28, -252, 495, -364, 120, -18, 1, -36, 462, -1287, 1365, -680, 171, -21, 1, 45, -792, 3003, -4368, 3060, -1140, 231, -24, 1, -55, 1287, -6435, 12376, -11628, 5985, -1771, 300, -27, 1

Offset: 0

Author

Paul Barry, Feb 04 2007

Keywords

• easy
• sign
• tabl

Comments

The matrix inverse of the convolution triangle of A001764 (number of ternary trees). - Peter Luschny, Oct 09 2022

Examples

			Triangle begins
    1;
   -3,     1;
    6,    -6,     1;
  -10,    21,    -9,      1;
   15,   -56,    45,    -12,      1;
  -21,   126,  -165,     78,    -15,      1;
   28,  -252,   495,   -364,    120,    -18,     1;
  -36,   462, -1287,   1365,   -680,    171,   -21,     1;
   45,  -792,  3003,  -4368,   3060,  -1140,   231,   -24,   1;
  -55,  1287, -6435,  12376, -11628,   5985, -1771,   300, -27,   1;
   66, -2002, 12870, -31824,  38760, -26334, 10626, -2600, 378, -30, 1;
		

Links

• G. C. Greubel, Rows n=0..100 of triangle, flattened

Crossrefs

Inverse is A127898.

Alternating sign version of A127893.

Cf. A001764, A095263, A127896.

Programs

Magma

[(-1)^(n-k)*Binomial(n+2*k+2, n-k): k in [0..n], n in [0..10]]; // G. C. Greubel, Apr 29 2018

Maple

# Uses function InvPMatrix from A357585. Adds column 1, 0, 0, ... to the left.
InvPMatrix(10, n -> binomial(3*n, n)/(2*n+1)); # Peter Luschny, Oct 09 2022

Mathematica

Table[(-1)^(n-k)*Binomial[n+2*k+2, n-k], {n,0,10}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 29 2018 *)

PARI

for(n=0, 10, for(k=0,n, print1((-1)^(n-k)*binomial(n+2*k+2, n-k), ", "))) \\ G. C. Greubel, Apr 29 2018

Sage

flatten([[(-1)^(n-k)*binomial(n+2*k+2, n-k) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Apr 16 2021

Formula

T(n, k) = (-1)^(n-k)*binomial(n +2*k +2, n-k).

Sum_{k=0..n} T(n, k) = A127896(n) (row sums).

Sum_{k=0..floor(n/2)} T(n-k, k) = (-1)^n*A095263(n) (diagonal sums).

Extensions

Terms a(50) onward added by G. C. Greubel, Apr 29 2018