A123876 Riordan array (1/(1+2*x), x*(1+x)/(1+2*x)^2).

1, -2, 1, 4, -5, 1, -8, 18, -8, 1, 16, -56, 41, -11, 1, -32, 160, -170, 73, -14, 1, 64, -432, 620, -377, 114, -17, 1, -128, 1120, -2072, 1666, -704, 164, -20, 1, 256, -2816, 6496, -6608, 3649, -1178, 223, -23, 1, -512, 6912, -19392, 24192, -16722, 7001, -1826, 291, -26, 1

Offset: 0

Author

Paul Barry, Oct 16 2006

Keywords

• easy
• sign
• tabl

Comments

Inverse of A116395.

Row sums are A123877.

Diagonal sums are (-1)^n*A085810(n).

Unsigned version is A114164.

Examples

			Triangle begins
    1;
   -2,   1;
    4,  -5,    1;
   -8,  18,   -8,   1;
   16, -56,   41, -11,   1;
  -32, 160, -170,  73, -14, 1;
		

Links

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

Crossrefs

Cf. A085810, A114164, A116395, A123877.

Programs

GAP

Flat(List([0..12], n-> List([0..n], k-> (-1)^(n-k)*Sum([0..n], j-> 2^(n-j)*Binomial(k,j-k)*Binomial(n,j) ))));

Magma

[(-1)^(n-k)*(&+[2^(n-j)*Binomial(k,j-k)*Binomial(n,j): j in [0..n]]): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 08 2019

Mathematica

Table[(-1)^(n-k)*Sum[2^(n-j)*Binomial[k,j-k]*Binomial[n,j], {j,0,n}], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 08 2019 *)

PARI

T(n,k) = b=binomial; (-1)^(n-k)*sum(j=0,n, 2^(n-j)*b(k,j-k)* b(n,j)); \\ G. C. Greubel, Aug 08 2019

Sage

b=binomial;
[[(-1)^(n-k)*sum(2^(n-j)*b(k,j-k)*b(n,j) for j in (0..n)) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 08 2019

Formula

Number triangle T(n,k) = (-1)^(n-k)*Sum_{j=0..n} C(k,j-k)*C(n,j)*2^(n-j).

T(n,k) = T(n-1,k-1) - 4*T(n-1,k) + T(n-2,k-1) - 4*T(n-2,k), T(0,0) = T(1,1) = 1, T(1,0) = -2, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Jan 18 2014

Extensions

More terms added by G. C. Greubel, Aug 08 2019