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A124576
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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 (1,4,4,...) and super- and subdiagonals (1,1,1,...).
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25
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1, 1, 1, 2, 5, 1, 7, 23, 9, 1, 30, 108, 60, 13, 1, 138, 522, 361, 113, 17, 1, 660, 2587, 2079, 830, 182, 21, 1, 3247, 13087, 11733, 5581, 1579, 267, 25, 1, 16334, 67328, 65600, 35636, 12164, 2672, 368, 29, 1, 83662, 351246, 365364, 220308, 86964, 23220, 4173
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,4
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COMMENTS
| 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)=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
| Sum{k, 0<=k<=n}T(n,k)*(4*k+1)=6^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 27 2007
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EXAMPLE
| Row 3 is (2,5,1) because M[3]=[1,1,0;1,4,1;0,1,4] and M[3]^2=[2,5,1;5,18,8;1,8,17].
Triangle starts:
1;
1, 1;
2, 5, 1;
7, 23, 9, 1;
30, 108, 60, 13, 1;
138, 522, 361, 113, 17, 1;
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MAPLE
| with(linalg): m:=proc(i, j) if i=1 and j=1 then 1 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; 1, 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. A124575, A124574, A052179.
Sequence in context: A060789 A134570 A019510 * A174815 A021401 A199868
Adjacent sequences: A124573 A124574 A124575 * A124577 A124578 A124579
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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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