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A137156
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Matrix inverse of triangle A137153(n,k) = C(2^k+n-k-1, n-k), read by rows.
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6
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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
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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0,5
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COMMENTS
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Unsigned column 0 = A001192, number of full sets of size n.
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LINKS
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FORMULA
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G.f. of column k: 1 = Sum_{n>=0} T(n+k,k)*x^n/(1-x)^(2^(n+k)).
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EXAMPLE
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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; ...
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PROG
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(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(""))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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