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A165621
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Riordan array (c(x^2)*(1+xc(x^2)), xc(x^2)).
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1
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1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 3, 1, 1, 5, 5, 4, 4, 1, 1, 5, 9, 9, 5, 5, 1, 1, 14, 14, 14, 14, 6, 6, 1, 1, 14, 28, 28, 20, 20, 7, 7, 1, 1, 42, 42, 48, 48, 27, 27, 8, 8, 1, 1, 42, 90, 90, 75, 75, 35, 35, 9, 9, 1, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,7
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COMMENTS
| Inverse of A165620. Row sums are A001405(n+1). Diagonal sums are A026008.
Factors as (1+xc(x^2),x)*(c(x^2),xc(x^2)). Transforms (-2)^n to a sequence with Hankel transform F(2n+1).
In general, the Hankel transform of r^n by this matrix with have a Hankel transform with g.f. (1-x)/(1+(r-1)x+x^2).
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FORMULA
| Number triangle T(n,k)=sum{j=0..n, b(n-j)*sum{i=0..k, (-1)^(k-i)*C(k,i)*sum{m=0..i, C(i,m)*(C(i-m,m+k)-C(i-m,i+k+2))}}}
where b(n) is the sequence beginning with 1 followed by the aerated Catalan numbers: 1,1,0,1,0,2,0,5,0,14,...
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EXAMPLE
| Triangle begins
1,
1, 1,
1, 1, 1,
2, 2, 1, 1,
2, 3, 3, 1, 1,
5, 5, 4, 4, 1, 1,
5, 9, 9, 5, 5, 1, 1,
14, 14, 14, 14, 6, 6, 1, 1,
14, 28, 28, 20, 20, 7, 7, 1, 1,
42, 42, 48, 48, 27, 27, 8, 8, 1, 1
The production arry of this matrix begins
1, 1,
0, 0, 1,
1, 1, 0, 1,
-1, 0, 1, 0, 1,
1, 0, 0, 1, 0, 1,
-1, 0, 0, 0, 1, 0, 1,
1, 0, 0, 0, 0, 1, 0, 1,
-1, 0, 0, 0, 0, 0, 1, 0, 1,
1, 0, 0, 0, 0, 0, 0, 1, 0, 1
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CROSSREFS
| Sequence in context: A066030 A025863 A136605 * A004739 A156282 A203776
Adjacent sequences: A165618 A165619 A165620 * A165622 A165623 A165624
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Sep 22 2009
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