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

1, -1, 1, 1, -2, 1, -2, 5, -4, 1, 9, -24, 22, -8, 1, -88, 239, -228, 92, -16, 1, 1802, -4920, 4749, -1976, 376, -32, 1, -75598, 206727, -200240, 84086, -16432, 1520, -64, 1, 6421599, -17568408, 17034964, -7173240, 1413084, -133984, 6112, -128, 1

Offset

0,5

Comments

Unsigned column 0 = A001192, number of full sets of size n.

Links

Table of n, a(n) for n=0..44.

Formula

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

Example

Triangle begins:
        1;
       -1,         1;
        1,        -2,        1;
       -2,         5,       -4,        1;
        9,       -24,       22,       -8,       1;
      -88,       239,     -228,       92,     -16,       1;
     1802,     -4920,     4749,    -1976,     376,     -32,    1;
   -75598,    206727,  -200240,    84086,  -16432,    1520,  -64,    1;
  6421599, -17568408, 17034964, -7173240, 1413084, -133984, 6112, -128, 1;
  ...

Prog

(PARI) /* As matrix inverse of A137153: */
{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])}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(PARI) /* Using the g.f.: */
{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)))}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A137153 (matrix inverse); unsigned columns: A001192, A137157, A137158, A137159; unsigned row sums: A137160.

Sequence in context: A337991 A104560 A121435 * A136457 A375048 A380246

Adjacent sequences: A137153 A137154 A137155 * A137157 A137158 A137159

Keyword

sign,tabl

Author

Paul D. Hanna, Jan 24 2008

Status

approved