cp's OEIS Frontend

A137156 Matrix inverse of triangle A137153(n,k) = C(2^k+n-k-1, n-k), read by rows.

%I A137156 #11 May 17 2025 03:11:40
%S A137156 1,-1,1,1,-2,1,-2,5,-4,1,9,-24,22,-8,1,-88,239,-228,92,-16,1,1802,
%T A137156 -4920,4749,-1976,376,-32,1,-75598,206727,-200240,84086,-16432,1520,
%U A137156 -64,1,6421599,-17568408,17034964,-7173240,1413084,-133984,6112,-128,1
%N A137156 Matrix inverse of triangle A137153(n,k) = C(2^k+n-k-1, n-k), read by rows.
%C A137156 Unsigned column 0 = A001192, number of full sets of size n.
%F A137156 G.f. of column k: 1 = Sum_{n>=0} T(n+k,k)*x^n/(1-x)^(2^(n+k)).
%e A137156 Triangle begins:
%e A137156         1;
%e A137156        -1,         1;
%e A137156         1,        -2,        1;
%e A137156        -2,         5,       -4,        1;
%e A137156         9,       -24,       22,       -8,       1;
%e A137156       -88,       239,     -228,       92,     -16,       1;
%e A137156      1802,     -4920,     4749,    -1976,     376,     -32,    1;
%e A137156    -75598,    206727,  -200240,    84086,  -16432,    1520,  -64,    1;
%e A137156   6421599, -17568408, 17034964, -7173240, 1413084, -133984, 6112, -128, 1;
%e A137156   ...
%o A137156 (PARI) /* As matrix inverse of A137153: */
%o A137156 {T(n,k) = local(M=matrix(n+1,n+1,r,c,if(r>=c,binomial(2^(c-1)+r-c-1,r-c)))); if(n<k||k<0,0,(M^-1)[n+1,k+1])}
%o A137156 for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
%o A137156 (PARI) /* Using the g.f.: */
%o A137156 {T(n,k) = if(n<k||k<0,0,if(n==k,1,polcoeff(1-sum(j=0, n-k-1, T(j+k,k)*x^j/(1-x+x*O(x^(n-k)))^(2^(j+k))),n-k)))}
%o A137156 for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
%Y A137156 Cf. A137153 (matrix inverse); unsigned columns: A001192, A137157, A137158, A137159; unsigned row sums: A137160.
%K A137156 sign,tabl
%O A137156 0,5
%A A137156 _Paul D. Hanna_, Jan 24 2008