|
|
A135950
|
|
Matrix inverse of triangle A022166.
|
|
11
|
|
|
1, -1, 1, 2, -3, 1, -8, 14, -7, 1, 64, -120, 70, -15, 1, -1024, 1984, -1240, 310, -31, 1, 32768, -64512, 41664, -11160, 1302, -63, 1, -2097152, 4161536, -2731008, 755904, -94488, 5334, -127, 1, 268435456, -534773760, 353730560, -99486720, 12850368, -777240, 21590, -255, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,4
|
|
COMMENTS
|
A022166 is the triangle of Gaussian binomial coefficients [n,k] for q = 2.
The coefficient [x^k] of Product_{i=1..n} (x-2^(i-1)). - Roger L. Bagula, Mar 20 2009
Triangle T(n,k), 0 <= k <= n, read by rows given by (-1, 1-q, -q^2, q-q^3, -q^4, q^2-q^5, -q^6, q^3-q^7, -q^8, ...) DELTA (1, 0, q, 0, q^2, 0, q^3, 0, q^4, 0, ...) (for q = 2) = (-1, -1, -4, -6, -16, -28, -64, -120, -256, ...) DELTA (1, 0, 2, 0, 4, 0, 8, 0, 16, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Oct 20 2013
|
|
LINKS
|
|
|
FORMULA
|
Unsigned column 0 equals A006125(n) = 2^(n*(n-1)/2). Unsigned column 1 equals A127850(n) = (2^n-1)*2^(n*(n-1)/2)/2^(n-1). Row sums equal 0^n. Unsigned row sums equal A028361(n) = Product_{k=0..n} (1+2^k).
|
|
EXAMPLE
|
Triangle begins:
1;
-1, 1;
2, -3, 1;
-8, 14, -7, 1;
64, -120, 70, -15, 1;
-1024, 1984, -1240, 310, -31, 1;
32768, -64512, 41664, -11160, 1302, -63, 1;
-2097152, 4161536, -2731008, 755904, -94488, 5334, -127, 1; ...
|
|
MATHEMATICA
|
max = 9; M = Table[QBinomial[n, k, 2], {n, 0, max}, {k, 0, max}] // Inverse; Table[M[[n, k]], {n, 1, max+1}, {k, 1, n}] // Flatten (* Jean-François Alcover, Apr 08 2016 *)
p[x_, n_, q_] := (-1)^n*q^Binomial[n, 2]*QPochhammer[x, 1/q, n];
Table[CoefficientList[Series[p[x, n, 2], {x, 0, n}], x], {n, 0, 10}]// Flatten (* G. C. Greubel, Apr 15 2019 *)
|
|
PROG
|
(PARI) T(n, k)=local(q=2, A=matrix(n+1, n+1, n, k, if(n>=k, if(n==1 || k==1, 1, prod(j=n-k+1, n-1, 1-q^j)/prod(j=1, k-1, 1-q^j))))^-1); A[n+1, k+1]
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|