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A099097
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Riordan array (1,3+x).
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3
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1, 0, 3, 0, 1, 9, 0, 0, 6, 27, 0, 0, 1, 27, 81, 0, 0, 0, 9, 108, 243, 0, 0, 0, 1, 54, 405, 729, 0, 0, 0, 0, 12, 270, 1458, 2187, 0, 0, 0, 0, 1, 90, 1215, 5103, 6561, 0, 0, 0, 0, 0, 15, 540, 5103, 17496, 19683, 0, 0, 0, 0, 0, 1, 135, 2835, 20412, 59049, 59049, 0, 0, 0, 0, 0, 0, 18
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Row sums are A006190(n+1). Diagonal sums are A052931. The Riordan array (1,s+tx) defines T(n,k)=binomial(k,n-k)s^k(t/s)^(n-k). The row sums satisfy a(n)=s*a(n-1)+t*a(n-2) and the diagonal sums satisfy a(n)=s*a(n-2)+t*a(n-3).
Triangle T(n,k), 0<=k<=n, read by rows given by [0, 1/3, -1/3, 0, 0, 0, 0, 0, ...] DELTA [3, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 10 2008]
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FORMULA
| Number triangle T(n, k)=binomial(k, n-k)*3^k*(1/3)^(n-k); Columns have g.f. (3x+x^2)^k.
G.f.: 1/(1-3y*x-y*x^2). - From DELEHAM Philippe, Nov 21 2011
Sum_{k, 0<=k<=n} T(n,k)*x^k = A000007(n), A006190(n+1), A1350330(n+1), A181353(n+1) for x = 0,1,2,3 respectively. - From DELEHAM Philippe, Nov 21 2011
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EXAMPLE
| Rows begin {1}, {0,3}, {0,1,9}, {0,0,6,27}, {0,0,1,27,81},...
Triangle begins :
1
0, 3
0, 1, 9
0, 0, 6, 27
0, 0, 1, 27, 81
0, 0, 0, 9, 108, 243
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CROSSREFS
| Cf. A027465.
Sequence in context: A020816 A174860 A157391 * A152150 A136239 A058175
Adjacent sequences: A099094 A099095 A099096 * A099098 A099099 A099100
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Sep 25 2004
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