A172400 - OEIS

%I #2 Mar 30 2012 18:37:21

%S 1,1,1,2,1,1,6,3,1,1,32,16,5,1,1,332,166,51,9,1,1,6928,3464,1059,181,

%T 17,1,1,292334,146167,44620,7557,681,33,1,1,24875760,12437880,3795202,

%U 641035,57097,2641,65,1,1,4254812880,2127406440,649054326,109540639

%N G.f.: 1/(1-x) = (1-x*y) * Sum_{k>=0} Sum_{n>=k} T(n,k)*x^n*y^k/(1+x)^(2^n-2^k).

%F Unsigned column 0 of matrix inverse forms A001192, which is the number of full sets of size n.

%e Triangle begins:

%e 1;

%e 1, 1;

%e 2, 1, 1;

%e 6, 3, 1, 1;

%e 32, 16, 5, 1, 1;

%e 332, 166, 51, 9, 1, 1;

%e 6928, 3464, 1059, 181, 17, 1, 1;

%e 292334, 146167, 44620, 7557, 681, 33, 1, 1;

%e 24875760, 12437880, 3795202, 641035, 57097, 2641, 65, 1, 1;

%e 4254812880, 2127406440, 649054326, 109540639, 9723237, 443921, 10401, 129, 1, 1; ...

%e Matrix inverse of this triangle begins:

%e 1;

%e -1,1;

%e -1,-1,1;

%e -2,-2,-1,1;

%e -9,-9,-4,-1,1;

%e -88,-88,-38,-8,-1,1;

%e -1802,-1802,-772,-156,-16,-1,1;

%e -75598,-75598,-32313,-6456,-632,-32,-1,1; ...

%e in which unsigned column 0 = A001192, number of full sets of size n.

%o (PARI) {T(n,k)=if(n==k,1,polcoeff(-(1-x)*sum(m=0,n-k-1,T(m+k,k)*x^m/(1+x +x*O(x^n))^(2^(m+k)-2^k)),n-k))}

%o (PARI) {T(n,k)=local(M,N); M=matrix(n+1,n+1,r,c,if(r>=c,polcoeff(1/(1-x+O(x^(r-c+1)))^1*(1+x)^(2^(r-1)-2^(c-1)),r-c))); N=matrix(n+1,n+1,r,c,if(r>=c,polcoeff(1/(1-x+O(x^(r-c+1)))^2*(1+x)^(2^(r-1)-2^(c-1)),r-c))); (M^-1*N)[n+1,k+1]}

%Y Cf. A001192, columns: A172401, A172402, A172403.

%K nonn,tabl

%O 0,4

%A _Paul D. Hanna_, Feb 01 2010