A188481 Riordan matrix (1/(1-4x),(1-sqrt(1-4x))/(2*sqrt(1-4x))).

1, 4, 1, 16, 7, 1, 64, 38, 10, 1, 256, 187, 69, 13, 1, 1024, 874, 406, 109, 16, 1, 4096, 3958, 2186, 748, 158, 19, 1, 16384, 17548, 11124, 4570, 1240, 216, 22, 1, 65536, 76627, 54445, 25879, 8485, 1909, 283, 25, 1, 262144, 330818, 259006, 138917, 52984, 14471, 2782, 359, 28, 1

Offset: 0

Author

Emanuele Munarini, Apr 01 2011

Keywords

• nonn
• easy
• tabl

Comments

Row sums = A141223;

Diagonal sums = A188482;

Inverse matrix: (1/(1+2x)^2, x(1+x)/(1+2x)^2).

Examples

			Triangle begins:
      1;
      4,     1;
     16,     7,     1;
     64,    38,    10,     1;
    256,   187,    69,    13,    1;
   1024,   874,   406,   109,   16,    1;
   4096,  3958,  2186,   748,  158,   19,   1;
  16384, 17548, 11124,  4570, 1240,  216,  22,  1;
  65536, 76627, 54445, 25879, 8485, 1909, 283, 25, 1;
		

Crossrefs

Cf. A141223, A188482.

Programs

Mathematica

Flatten[Table[Sum[Binomial[n+i,n]Binomial[n-i,k]2^(n-k-i),{i,0,n-k}],{n,0,8},{k,0,8}]]

Maxima

create_list(sum(binomial(n+i,n)*binomial(n-i,k)*2^(n-k-i),i,0,n-k),n,0,8,k,0,n);

Formula

T(n,k) = [x^n] ((1-sqrt(1-4*x))/(2*sqrt(1-4*x)))^k/(1-4*x).

Recurrence: T(n+1,k+1) = T(n,k) + 3*T(n,k-1) + T(n,k-2) - T(n,k-3) + T(n,k-4) - T(n,k-5) + ...

Extensions

Comment corrected by Philippe Deléham, Jan 22 2014