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A124574
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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 (3,4,4,...) and super- and subdiagonals (1,1,1,...).
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25
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1, 3, 1, 10, 7, 1, 37, 39, 11, 1, 150, 204, 84, 15, 1, 654, 1050, 555, 145, 19, 1, 3012, 5409, 3415, 1154, 222, 23, 1, 14445, 28063, 20223, 8253, 2065, 315, 27, 1, 71398, 146920, 117208, 55300, 16828, 3352, 424, 31, 1, 361114, 776286, 671052, 355236
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Column 1 yields A064613 (2nd binomial transform of the Catalan sequence A000108). Row sums yield A081671.
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)=3*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) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 27 2007
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)=3*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
6^n = ((n+1)-th row terms) dot (first n+1 odd integers). Example: 6^4 = 1296 = (150, 204, 84, 15, 1) dot (1, 3, 5, 7, 9) = (150 + 612 + 420 + 105 + 9)= 1296. - Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 15 2011
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FORMULA
| Sum_{k, 0<=k<=n}(-1)^(n-k)*T(n,k)=(-2)^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 27 2007
Sum_{k, 0<=k<=n}T(n,k)*(2*k+1)=6^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 27 2007
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EXAMPLE
| Row 4 is (37,39,11,1) because M[4]= [3,1,0,0;1,4,1,0;0,1,4,1;0,0,1,4] and M[4]^3=[37,39,11,1; 39, 87, 51, 12; 11, 51, 88, 50; 1, 12, 50, 76].
Triangle starts:
1;
3, 1
10, 7, 1;
37, 39, 11, 1
150, 204, 84, 15, 1;
654, 1050, 555, 145, 19, 1;
Production matrix begins
3, 1
1, 4, 1
0, 1, 4, 1
0, 0, 1, 4, 1
0, 0, 0, 1, 4, 1
0, 0, 0, 0, 1, 4, 1
0, 0, 0, 0, 0, 1, 4, 1
0, 0, 0, 0, 0, 0, 1, 4, 1
0, 0, 0, 0, 0, 0, 0, 1, 4, 1
- From DELEHAM Philippe, Nov 07 2011
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MAPLE
| with(linalg): m:=proc(i, j) if i=1 and j=1 then 3 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; 3, 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. A000108, A081671, A124575, A124576, A052179, A064613.
Sequence in context: A116384 A117207 A046658 * A052964 A084178 A195812
Adjacent sequences: A124571 A124572 A124573 * A124575 A124576 A124577
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KEYWORD
| nonn,tabl
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AUTHOR
| Gary W. Adamson & Roger L. Bagula (qntmpkt(AT)yahoo.com), Nov 04 2006
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), Dec 04 2006
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