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A099089
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Riordan array (1,2+x).
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4
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1, 0, 2, 0, 1, 4, 0, 0, 4, 8, 0, 0, 1, 12, 16, 0, 0, 0, 6, 32, 32, 0, 0, 0, 1, 24, 80, 64, 0, 0, 0, 0, 8, 80, 192, 128, 0, 0, 0, 0, 1, 40, 240, 448, 256, 0, 0, 0, 0, 0, 10, 160, 672, 1024, 512, 0, 0, 0, 0, 0, 1, 60, 560, 1792, 2304, 1024, 0, 0, 0, 0, 0, 0, 12, 280, 1792, 4608, 5120, 2048
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Row sums are A000129. Diagonal sums are A008346. 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, 1/2, -1/2, 0, 0, 0, 0, ...] DELTA [2, 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)2^k(1/2)^(n-k) Columns have g.f. (2x+x^2)^k.
G.f.: 1/(1-2y*x-y*x^2). - From DELEHAM Philippe, Nov 20 2011
Sum_ {k, 0<=k<=n} T(n,k)*x^k = A000007(n), A000129(n+1), A090017(n+1), A090018(n), A190510(n+1), A190955(n+1) for x = 0,1,2,3,4,5 respectively. - From DELEHAM Philippe, Nov 20 2011
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EXAMPLE
| Rows begin {1}, {0,2}, {0,1,4}, {0,0,4,8}, {0,0,1,12,16},...
Triangle begins :
1
0, 2
0, 1, 4
0, 0, 4, 8
0, 0, 1, 12, 16
0, 0, 0, 6, 32, 32
0, 0, 0, 1, 24, 80, 64
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CROSSREFS
| Cf. A053118, A008312.
Sequence in context: A072737 A061290 A099096 * A121298 A121462 A131487
Adjacent sequences: A099086 A099087 A099088 * A099090 A099091 A099092
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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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