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A081578
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Pascal-(1,3,1) array.
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9
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1, 1, 1, 1, 5, 1, 1, 9, 9, 1, 1, 13, 33, 13, 1, 1, 17, 73, 73, 17, 1, 1, 21, 129, 245, 129, 21, 1, 1, 25, 201, 593, 593, 201, 25, 1, 1, 29, 289, 1181, 1921, 1181, 289, 29, 1, 1, 33, 393, 2073, 4881, 4881, 2073, 393, 33, 1, 1, 37, 513, 3333, 10497, 15525, 10497, 3333, 513
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| One of a family of Pascal-like arrays. A007318 is equivalent to the (1,0,1)-array. A008288 is equivalent to the (1,1,1)-array. Rows include A016813, A081585, A081586. Coefficients of the row polynomials in the Newton basis are given by A013611.
As a number triangle, this is the Riordan array (1/(1-x), x(1+3x)/(1-x)). It has row sums A015518(n+1) and diagonal sums A103143. - Paul Barry (pbarry(AT)wit.ie), Jan 24 2005
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REFERENCES
| Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
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FORMULA
| Square array T(n, k) defined by T(n, 0)=T(0, k)=1, T(n, k)=T(n, k-1)+3T(n-1, k-1)+T(n-1, k). Rows are the expansions of (1+3x)^k/(1-x)^(k+1)
T(n,k)=sum{j=0..n, C(k,j-k)*C(n+k-j,k)*3^(j-k)}; - Paul Barry (pbarry(AT)wit.ie), Oct 23 2006
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EXAMPLE
| Rows begin
1 1 1 1 1 ...
1 5 9 13 17 ...
1 9 33 73 129 ...
1 13 73 245 593 ...
1 17 129 593 1921 ...
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CROSSREFS
| Cf. A081577, A081579, A081580.
Sequence in context: A183450 A131061 A157169 * A184883 A188461 A188474
Adjacent sequences: A081575 A081576 A081577 * A081579 A081580 A081581
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Mar 23 2003
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