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A124733
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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 (2,3,3,...) and super- and subdiagonals (1,1,1,...).
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27
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1, 2, 1, 5, 5, 1, 15, 21, 8, 1, 51, 86, 46, 11, 1, 188, 355, 235, 80, 14, 1, 731, 1488, 1140, 489, 123, 17, 1, 2950, 6335, 5397, 2730, 875, 175, 20, 1, 12235, 27352, 25256, 14462, 5530, 1420, 236, 23, 1, 51822, 119547, 117582, 74172, 32472, 10026, 2151, 306, 26, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| 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)+3*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)=2*T(n-1,0)+T(n-1,1), T(n,k)=T(n-1,k-1)+3*T(n-1,k)+T(n-1,k+1) for k>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 27 2007
Equals A007318*A039599 (when written as lower triangular matrix) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 16 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
5^n = (n-th row terms) dot (first n+1 odd integers). Example: 5^4 = 625 = (51, 86, 46, 11, 1) dot (1, 3, 5, 7, 9) = (51 + 258 + 230 + 77 + 9) = 625. [From Gary W. Adamson, Jun 13 2011]
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FORMULA
| Sum_{k, 0<=k<=n}(-1)^(n-k)*T(n,k)=(-1)^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 27 2007
Sum_{k, 0<=k<=n}T(n,k)*(2*k+1)=5^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 27 2007
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EXAMPLE
| Row 3 is (5,5,1) because M[3]=[2,1,0;1,3,1;0,1,3] and M[3]^2=[5,5,1;5,11,6;1,6,10].
Triangle starts:
1;
2, 1;
5, 5, 1;
15, 21, 8, 1;
51, 86, 46, 11, 1;
188, 355, 235, 80, 14, 1;
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MAPLE
| with(linalg): m:=proc(i, j) if i=1 and j=1 then 2 elif i=j then 3 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. A110877, A091965, A007317, A007317, A026375.
Sequence in context: A060920 A107842 A126216 * A137597 A059340 A204119
Adjacent sequences: A124730 A124731 A124732 * A124734 A124735 A124736
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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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