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A104967
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Matrix inverse of triangle A104219, read by rows, where A104219(n,k) equals the number of Schroeder paths of length 2n having k peaks at height 1.
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4
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1, -1, 1, -1, -2, 1, -1, -1, -3, 1, -1, 0, 0, -4, 1, -1, 1, 2, 2, -5, 1, -1, 2, 3, 4, 5, -6, 1, -1, 3, 3, 3, 5, 9, -7, 1, -1, 4, 2, 0, 0, 4, 14, -8, 1, -1, 5, 0, -4, -6, -6, 0, 20, -9, 1, -1, 6, -3, -8, -10, -12, -14, -8, 27, -10, 1, -1, 7, -7, -11, -10, -10, -14, -22, -21, 35, -11, 1, -1, 8, -12, -12, -5, 0, 0, -8, -27, -40, 44, -12, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Row sums equal A090132 with odd-indexed terms negated. Absolute row sums form A104968. Row sums of squared terms gives A104969.
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FORMULA
| G.f.: A(x, y) = (1-2*x)/(1-x - x*y*(1-2*x)).
T(n,k)=sum(i=0..n-k, (-2)^i*binomial(k+1,i)*binomial(n-i,k)). [From Vladimir Kruchinin, Nov 02 2011]
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EXAMPLE
| Triangle begins:
1;
-1,1;
-1,-2,1;
-1,-1,-3,1;
-1,0,0,-4,1;
-1,1,2,2,-5,1;
-1,2,3,4,5,-6,1;
-1,3,3,3,5,9,-7,1;
-1,4,2,0,0,4,14,-8,1;
-1,5,0,-4,-6,-6,0,20,-9,1; ...
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PROG
| (PARI) {T(n, k)=local(X=x+x*O(x^n), Y=y+y*O(y^k)); polcoeff(polcoeff((1-2*X)/(1-X-X*Y*(1-2*X)), n, x), k, y)}
(Maxima) T(n, k):=sum((-2)^i*binomial(k+1, i)*binomial(n-i, k), i, 0, n-k); [From Vladimir Kruchinin, Nov 02 2011]
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CROSSREFS
| Cf. A090132, A104969, A104969.
Sequence in context: A073266 A125692 A128258 * A098495 A175432 A204118
Adjacent sequences: A104964 A104965 A104966 * A104968 A104969 A104970
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KEYWORD
| sign,tabl
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Mar 30 2005
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