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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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6
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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
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OFFSET
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0,5
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COMMENTS
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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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LINKS
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FORMULA
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G.f.: A(x, y) = (1-2*x)/(1-x - x*y*(1-2*x)).
Sum_{k=0..n} T(n, k) = (-1)^n*A090132(n).
Sum_{k=0..n} abs(T(n, k)) = A104968(n).
Sum_{k=0..n} T(n, k)^2 = A104969(n).
T(n,k) = Sum_{i=0..n-k} (-2)^i*binomial(k+1,i)*binomial(n-i,k). - Vladimir Kruchinin, Nov 02 2011
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EXAMPLE
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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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MAPLE
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A104967:= (n, k)-> add( (-2)^j*binomial(k+1, j)*binomial(n-j, k), j=0..n-k);
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MATHEMATICA
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T[n_, k_]:= T[n, k]= Which[k==n, 1, k==0, 0, True, T[n-1, k-1] - Sum[T[n-i, k-1], {i, 2, n-k+1}]];
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PROG
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(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)}
for(n=0, 16, for(k=0, n, print1(T(n, k), ", ")); print(""))
(Maxima) T(n, k):=sum((-2)^i*binomial(k+1, i)*binomial(n-i, k), i, 0, n-k); \\ Vladimir Kruchinin, Nov 02 2011
(Sage)
@cached_function
def prec(n, k):
if k==n: return 1
if k==0: return 0
return prec(n-1, k-1)-sum(prec(n-i, k-1) for i in (2..n-k+1))
return [prec(n, k) for k in (1..n)]
(Magma)
A104967:= func< n, k | (&+[(-2)^j*Binomial(k+1, j)*Binomial(n-j, k): j in [0..n-k]]) >;
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CROSSREFS
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Cf. A347171 (rows reversed, up to signs).
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KEYWORD
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AUTHOR
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STATUS
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approved
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