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A133367
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Triangle T(n,k) read by rows given by [2,1,2,1,2,1,2,1,2,1,2,1,...] DELTA [1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938 .
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0
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1, 2, 1, 6, 5, 1, 22, 23, 8, 1, 90, 107, 49, 11, 1, 394, 509, 276, 84, 14, 1, 1806, 2473, 1505, 556, 128, 17, 1, 8558, 12235, 8100, 3429, 974, 181, 20, 1, 41586, 61463, 43393, 20355, 6713, 1557, 243, 23, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Triangle begins : 1 ; 2, 1 6, 5, 1 ; 22, 23, 8, 1 ; 90, 107, 49, 11, 1 ; 394, 509, 276, 84, 14, 1 ; 1806, 2473, 1505, 556, 128, 17, 1 ; 8558, 12235, 8100, 3429, 974, 181, 20, 1 ; 41586, 61463, 43393, 20355, 6713, 1557, 243, 23, 1 ; ...
Contribution from Paul Barry (pbarry(AT)wit.ie), Apr 28 2009: (Start)
Riordan array ((1-x-sqrt(1-6x+x^2))/(2x), (1-3x-sqrt(1-6x+x^2))/(4x)).
Inverse of Riordan array (1/(1+2x),x/(1+3x+2x^2)) (a signed version of A124237). (End)
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FORMULA
| T(0,0)=1 ; T(n,k)=0 if k<0 or if k>n ; T(n,0) = 2*T(n-1,0)+2*T(n-1,1) ; T(n,k) = T(n-1,k-1)+3*T(n-1,k)+2*T(n-1,k+1) for k>=1 .
G.f.: 1/(1-xy-2x-x^2(2+y)/(1-3x-2x^2/(1-3x-2x^2/(1-3x-2x^2/(1- ... (continued fraction) [From Paul Barry (pbarry(AT)wit.ie), Apr 28 2009]
Sum_{k, k>=0} T(m,k)*T(n,k)*2^k = T(m+n,0) = A006318(m+n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jan 24 2010]
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EXAMPLE
| Contribution from Paul Barry (pbarry(AT)wit.ie), Apr 28 2009: (Start)
Triangle begins
1,
2, 1,
6, 5, 1,
22, 23, 8, 1,
90, 107, 49, 11, 1,
394, 509, 276, 84, 14, 1,
1806, 2473, 1505, 556, 128, 17, 1
Production matrix begins
2, 1,
2, 3, 1,
0, 2, 3, 1,
0, 0, 2, 3, 1,
0, 0, 0, 2, 3, 1,
0, 0, 0, 0, 2, 3, 1,
0, 0, 0, 0, 0, 2, 3, 1; (End)
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CROSSREFS
| Cf.: A006318, A000012, A016789.
Sequence in context: A159965 A116395 A159924 * A179456 A121576 A121575
Adjacent sequences: A133364 A133365 A133366 * A133368 A133369 A133370
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KEYWORD
| nonn,tabl
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AUTHOR
| Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 27 2007
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