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A124182
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A skewed version of triangular array A081277.
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9
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1, 0, 1, 0, 1, 2, 0, 0, 3, 4, 0, 0, 1, 8, 8, 0, 0, 0, 5, 20, 16, 0, 0, 0, 1, 18, 48, 32, 0, 0, 0, 0, 7, 56, 112, 64, 0, 0, 0, 0, 1, 32, 160, 256, 128, 0, 0, 0, 0, 0, 9, 120, 432, 576, 256, 0, 0, 0, 0, 0, 1, 50, 400, 1120, 1280, 512
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,6
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COMMENTS
| Triangle T(n,k), 0<=k<=n, read by rows given by [0, 1, -1, 0, 0, 0, 0, 0, 0, ...] DELTA [1, 1, 0, 0, 0, 0, 0, 0, 0,...] where DELTA is the operator defined in A084938 . Falling diagonal sums in A052980.
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FORMULA
| T(0,0)=T(1,1)=1, T(n,k)=0 if n<k or if k<0,T(n,k)=T(n-2,k-1)+2*T(n-1,k-1) . Sum_{k, 0<=k<=n}x^k*T(n,k)= (-1)^n*A090965(n), (-1)^n*A084120(n), (-1)^n*A006012(n), A033999(n), A000007(n), A001333(n), A084059(n) for x= -4, -3, -2, -1, 0, 1, 2 respectively . Sum_{k, 0<=k<=[n/2]} T(n-k,k)= Fibonacci(n-1)= A000045(n-1).
Sum_{k, 0<=k<=n}T(n,k)*x^(n-k)= A000012(n), A011782(n), A001333(n), A026150(n), A046717(n), A084057(n), A002533(n), A083098(n), A084058(n), A003665(n), A002535(n), A133294(n), A090042(n), A125816(n), A133343(n), A133345(n), A120612(n), A133356(n), A125818(n) for x= -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17 respectively . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 26 2007
Sum_{k, 0<=k<=n}T(n,k)*(-x)^(n-k)= A011782(n), A000012(n), A146559(n), A087455(n), A138230(n), A006495(n), A138229(n) for x= 0,1,2,3,4,5,6 respectively. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 14 2008]
G.f.: (1-y*x)/(1-2y*x-y*x^2). - DELEHAM Philippe, Dec 04 2011
Sum_{k, 0<=k<=n} T(n,k)^2 = A002002(n) for n>0. - DELEHAM Philippe, Dec 04 2011
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EXAMPLE
| Triangle begins:
.1;
.0, 1;
.0, 1, 2;
.0, 0, 3, 4;
.0, 0, 1, 8, 8;
.0, 0, 0, 5, 20, 16;
.0, 0, 0, 1, 18, 48, 32;
.0, 0, 0, 0, 7, 56, 112, 64;
.0, 0, 0, 0, 1, 32, 160, 256, 128;
.0, 0, 0, 0, 0, 9, 120, 432, 576, 256;
.0, 0, 0, 0, 0, 1, 50, 400, 1120, 1280, 512;
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CROSSREFS
| Cf. A025192 (column sums). Diagonals include : A011782, A001792, A001793, A001794, A006974, A006975, A006976
Sequence in context: A050186 A074734 A174956 * A188429 A188430 A013585
Adjacent sequences: A124179 A124180 A124181 * A124183 A124184 A124185
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KEYWORD
| nonn,tabl
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AUTHOR
| Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Dec 05 2006
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