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A121314
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Triangle T(n,k), 0<=k<=n, read by rows given by [0, 1, 0, 0, 0, 0, ...] DELTA [1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
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6
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1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 5, 6, 1, 0, 1, 7, 15, 10, 1, 0, 1, 9, 28, 35, 15, 1, 0, 1, 11, 45, 84, 70, 21, 1, 0, 1, 13, 66, 165, 210, 126, 28, 1, 0, 1, 15, 91, 286, 495, 462, 210, 36, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,9
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COMMENTS
| A054142 with first diagonal 1, 0, 0, 0, 0, 0, 0, 0, ...
Mirror image of triangle in A165253.
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REFERENCES
| F. Yano and H. Yoshida, Some set partition statistics in non-crossing partitions and generating functions, Discr. Math., 307 (2007), 3147-3160.
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FORMULA
| T(0,0)=1; T(n,0)=0 for n>0 ; T(n+1,k+1)= binomial(2*n-k,k)for n>=0 and k>=0 .
Sum_{k, 0<=k<=n}T(n,k)*x^k = A001519(n), A047849(n), A165310(n), A165311(n), A165312(n), A165314(n), A165322(n), A165323(n), A165324(n) for x = 1,2,3,4,5,6,7,8,9 respectively. . Sum_{k, 0<=k<=n} 2^k*T(n,k) = (4^n+2)/3 . Sum_{k, 0<=k<=n} 2^(n-k)*T(n,k) = A001835(n).
Sum_{k, 0<=k<=n} 3^k*4^(n-k)*T(n,k) = A054879(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 26 2006
Sum_{k, 0<=k<=n}T(n,k)*(-1)^k*2^(3n-2k)=A143126(n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 31 2008]
Sum_{k, 0<=k<=n}T(n,k)*(-1)^k*3^(n-k)=A138340(n)/4^n . [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 01 2008]
G.f.: (1-(y+1)*x)/(1-(2y+1)*x+y^2*x^2). - From DELEHAM Philippe, Nov 01 2011
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EXAMPLE
| Triangle begins
1
0 1
0 1 1
0 1 3 1
0 1 5 6 1
0 1 7 15 10 1
0 1 9 28 35 15 1
0 1 11 45 84 70 21 1
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CROSSREFS
| Cf. A054142.
Sequence in context: A050143 A103495 A081719 * A119271 A125104 A098157
Adjacent sequences: A121311 A121312 A121313 * A121315 A121316 A121317
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KEYWORD
| nonn,tabl
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AUTHOR
| Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 25 2006
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