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

0,5

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

Links

Table of n, a(n) for n=0..42.

Example

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;

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 *)

Crossrefs

Sequence in context: A166346 A157640 A142458 * A259465 A176227 A340560

Adjacent sequences: A174525 A174526 A174527 * A174529 A174530 A174531

Keyword

nonn,tabl

Author

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

Status

approved