A135951 -id:A135951 - cp's OEIS frontend

A136097 a(n) = A135951(n) /[(2^(n+1)-1) * 2^(n*(n-1)/2)].

1, -1, 5, -93, 6477, -1733677, 1816333805, -7526310334829, 124031223014725741, -8152285307423733458541, 2140200604371078953284092525, -2245805993494514875022552272042605, 9423041917569791458584837551185555483245

Offset: 0

Paul D. Hanna, Dec 13 2007

sign

A135951 is the central terms of A135950; A135950 is the matrix inverse of A022166; A022166 is the triangle of Gaussian binomial coefficients [n,k] for q = 2.

Cf. A135951, A135950, A022166.

Table[(-1)^n QBinomial[2n, n, 2]/(2^(n+1) - 1), {n, 0, 20}] (* Vladimir Reshetnikov, Sep 16 2016 *)

a(n)=local(q=2,A=matrix(2*n+1,2*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[2*n+1,n+1]/( (q^(n+1)-1)/(q-1) * q^(n*(n-1)/2) )

Conjecture: the n-th central term of the matrix inverse of the triangle of Gaussian binomial coefficients in q is divisible by [(q^(n+1)-1)/(q-1) * q^(n*(n-1)/2)] for n>=0 and integer q > 1.

a(n) = (-1)^n * A015030(n) where A015030 is 2-Catalan numbers. - Michael Somos, Jan 10 2023

A135950 Matrix inverse of triangle A022166.

Original entry on oeis.org

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

Offset: 0

Paul D. Hanna, Dec 08 2007

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
Reversed rows of triangle A158474. - Werner Schulte, Apr 06 2019
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

			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; ...

G. C. Greubel, Rows n = 0..75 of triangle, flattened

Cf. A022166, A006125, A028361, A127850, A135951 (central terms), A158474.

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 *)

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]

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).
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]
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

cp's OEIS Frontend

A136097 a(n) = A135951(n) /[(2^(n+1)-1) * 2^(n*(n-1)/2)].

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