A174528 Triangle T(n,m) = 2*A022168(n,m) - binomial(n, m), 0 <= m <= n, read by rows.

1, 1, 1, 1, 8, 1, 1, 39, 39, 1, 1, 166, 708, 166, 1, 1, 677, 11584, 11584, 677, 1, 1, 2724, 186171, 753590, 186171, 2724, 1, 1, 10915, 2981685, 48417191, 48417191, 2981685, 10915, 1, 1, 43682, 47718190, 3101684114, 12443227012, 3101684114, 47718190

Offset: 0

Author

Roger L. Bagula and Gary W. Adamson, Mar 21 2010

Keywords

• nonn
• tabl

Comments

Row sums are 1, 2, 10, 80, 1042, 24524, 1131382, 102819584, 18742118986, 6775774063892, 4926666912583390, ... = 2*A006118(n) - 2^n.

This triangle essentially compares a Gaussian binomial equivalent to Pascal's triangle and Pascal's triangle itself. - Alonso del Arte, Nov 14 2011

Examples

			Triangle begins
  1;
  1,     1;
  1,     8,        1;
  1,    39,       39,          1;
  1,   166,      708,        166,           1;
  1,   677,    11584,      11584,         677,          1;
  1,  2724,   186171,     753590,      186171,       2724,        1;
  1, 10915,  2981685,   48417191,    48417191,    2981685,    10915,     1;
  1, 43682, 47718190, 3101684114, 12443227012, 3101684114, 47718190, 43682, 1;
		

Programs

Maple

A174528 := proc(n,k)
        2*A022168(n,k)-binomial(n,k) ;
end proc:
seq(seq(A174528(n,m),m=0..n),n=0..10) ; # R. J. Mathar, Nov 14 2011

Mathematica

c[n_, q_] = Product[1 - q^i, {i, 1, n}];
t[n_, m_, q_] = 2*c[n, q]/(c[m, q]*c[n - m, q]) - Binomial[n, m];
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 2, 12}]
(* alternate program *)
(* First run the program for A022168 to define gaussianBinom *)
Column[Table[2gaussianBinom[n, k, 4] - Binomial[n, k], {n, 0, 8}, {k, 0, n}], Center] (* Alonso del Arte, Nov 14 2011 *)