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A135950 Matrix inverse of triangle A022166.

%I A135950 #38 Jun 11 2024 01:39:52
%S A135950 1,-1,1,2,-3,1,-8,14,-7,1,64,-120,70,-15,1,-1024,1984,-1240,310,-31,1,
%T A135950 32768,-64512,41664,-11160,1302,-63,1,-2097152,4161536,-2731008,
%U A135950 755904,-94488,5334,-127,1,268435456,-534773760,353730560,-99486720,12850368,-777240,21590,-255,1
%N A135950 Matrix inverse of triangle A022166.
%C A135950 A022166 is the triangle of Gaussian binomial coefficients [n,k] for q = 2.
%C A135950 The coefficient [x^k] of Product_{i=1..n} (x-2^(i-1)). - _Roger L. Bagula_, Mar 20 2009
%C A135950 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
%C A135950 Reversed rows of triangle A158474. - _Werner Schulte_, Apr 06 2019
%C A135950 T(n,k) = Sum mu(0,U) where the sum is taken over the subspaces U of GF(2)^n having dimension n-k and mu is the Moebius function of the poset of all subspaces of GF(2)^n. - _Geoffrey Critzer_, Jun 02 2024
%H A135950 G. C. Greubel, <a href="/A135950/b135950.txt">Rows n = 0..75 of triangle, flattened</a>
%F A135950 Unsigned column 0 equals A006125(n) = 2^(n*(n-1)/2).
%F A135950 Unsigned column 1 equals A127850(n) = (2^n-1)*2^(n*(n-1)/2)/2^(n-1).
%F A135950 Row sums equal 0^n.
%F A135950 Unsigned row sums equal A028361(n) = Product_{k=0..n} (1+2^k).
%F A135950 T(n,k) = (-1)^(n-k) * A022166(n,k) * 2^binomial(n-k,2) for 0 <= k <= n. - _Werner Schulte_, Apr 06 2019 [corrected by _Werner Schulte_, Dec 27 2021]
%F A135950 Sum_{n>=0} Sum_{k=0..n} T(n,k)y^k*x^n/A005329(n) = e(y*x)/e(x) where e(x) = Sum_{n>=0} x^n/A005329(n). - _Geoffrey Critzer_, Jun 02 2024
%e A135950 Triangle begins:
%e A135950          1;
%e A135950         -1,       1;
%e A135950          2,      -3,        1;
%e A135950         -8,      14,       -7,      1;
%e A135950         64,    -120,       70,    -15,      1;
%e A135950      -1024,    1984,    -1240,    310,    -31,    1;
%e A135950      32768,  -64512,    41664, -11160,   1302,  -63,    1;
%e A135950   -2097152, 4161536, -2731008, 755904, -94488, 5334, -127, 1; ...
%t A135950 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 *)
%t A135950 p[x_, n_, q_] := (-1)^n*q^Binomial[n, 2]*QPochhammer[x, 1/q, n];
%t A135950 Table[CoefficientList[Series[p[x, n, 2], {x, 0, n}], x], {n, 0, 10}]// Flatten (* _G. C. Greubel_, Apr 15 2019 *)
%o A135950 (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]
%Y A135950 Cf. A022166, A006125, A028361, A127850, A135951 (central terms), A158474.
%K A135950 sign,tabl
%O A135950 0,4
%A A135950 _Paul D. Hanna_, Dec 08 2007