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A124575
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Triangle read by rows: row n is the first row of the matrix M[n]^(n-1), where M[n] is the n X n tridiagonal matrix with main diagonal (2,4,4,...) and super- and subdiagonals (1,1,1,...).
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29
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1, 2, 1, 5, 6, 1, 16, 30, 10, 1, 62, 146, 71, 14, 1, 270, 717, 444, 128, 18, 1, 1257, 3582, 2621, 974, 201, 22, 1, 6096, 18206, 15040, 6718, 1800, 290, 26, 1, 30398, 93960, 85084, 43712, 14208, 2986, 395, 30, 1, 154756, 491322, 478008, 274140, 103530
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Column k=0 yields A033543 (2nd binomial transform of the sequence A000957(n+1)). Row sums yield A133158. [Corrected by Philippe DELEHAM, Oct 24 2007, Dec 05 2009]
Triangle T(n,k), 0<=k<=n, defined by : T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=2*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+4*T(n-1,k)+T(n-1,k+1) for k>0.
Triangle T(n,k), 0<=k<=n, read by rows given by : T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=2*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+4*T(n-1,k)+T(n-1,k+1) for k>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 27 2007
This triangle belongs to the family of triangles defined by: T(0,0)=1, T(n,k)=0 if k<0 or if k>n, T(n,0)=x*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+y*T(n-1,k)+T(n-1,k+1) for k>=1 . Other triangles arise by choosing different values for (x,y): (0,0) -> A053121; (0,1) -> A089942; (0,2) -> A126093; (0,3) -> A126970; (1,0)-> A061554; (1,1) -> A064189; (1,2) -> A039599; (1,3) -> A110877; ((1,4) -> A124576; (2,0) -> A126075; (2,1) -> A038622; (2,2) -> A039598; (2,3) -> A124733; (2,4) -> A124575; (3,0) -> A126953; (3,1) -> A126954; (3,2) -> A111418; (3,3) -> A091965; (3,4) -> A124574; (4,3) -> A126791; (4,4) -> A052179; (4,5) -> A126331; (5,5) -> A125906 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 25 2007
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FORMULA
| T(n,k)=T(n-1,k-1)+4T(n-1,k)+T(n-1,k-1) for k>=2.
Sum_{k, 0<=k<=n}T(n,k)*(3*k+1)=6^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 27 2007
Sum_{k, k>=0} T(m,k)*T(n,k) = T(m+n,0)= A033543(m+n). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 22 2009]
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EXAMPLE
| Row 2 is (5,6,1) because M[3]= [2,1,0;1,4,1;0,1,4] and M[3]^2=[5,6,1;6,18,8;1,8,17].
Triangle starts:
1;
2, 1;
5, 6, 1;
16, 30, 10, 1;
62, 146, 71, 14, 1;
270, 717, 444, 128, 18, 1;
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MAPLE
| with(linalg): m:=proc(i, j) if i=1 and j=1 then 2 elif i=j then 4 elif abs(i-j)=1 then 1 else 0 fi end: for n from 3 to 11 do A[n]:=matrix(n, n, m): B[n]:=multiply(seq(A[n], i=1..n-1)) od: 1; 2, 1; for n from 3 to 11 do seq(B[n][1, j], j=1..n) od; # yields sequence in triangular form
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CROSSREFS
| Cf. A124576, A124574, A052179, A064613, A133158.
Sequence in context: A128567 A179455 A039810 * A178121 A113345 A078123
Adjacent sequences: A124572 A124573 A124574 * A124576 A124577 A124578
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KEYWORD
| nonn,tabl
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AUTHOR
| Gary W. Adamson & Roger L. Bagula (qntmpkt(AT)yahoo.com), Nov 05 2006
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Dec 04 2006
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