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A174528
Triangle T(n,m) = 2*A022168(n,m) - binomial(n, m), 0 <= m <= n, read by rows.
1
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
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
KEYWORD
nonn,tabl
AUTHOR
STATUS
approved