Revision History for A124575
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Showing entries 1-10
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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,...).
(history;
published version)
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#24 by Susanna Cuyler at Mon Jan 20 21:41:32 EST 2020
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#23 by Jon E. Schoenfield at Mon Jan 20 16:00:52 EST 2020
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#22 by Jon E. Schoenfield at Mon Jan 20 16:00:49 EST 2020
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| COMMENTS
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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 Deléham_, 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 byfrom 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 Deléham_, Sep 25 2007
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| FORMULA
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T(n,k) = T(n-1,k-1) + 4*T(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 Deléham_, Mar 27 2007
Sum_{k, k>=0} T(m,k)*T(n,k) = T(m+n,0)= ) = A033543(m+n). - Philippe Deléham, Nov 22 2009
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| EXAMPLE
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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;
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approved
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#21 by N. J. A. Sloane at Fri Mar 03 19:38:41 EST 2017
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#20 by G. C. Greubel at Fri Mar 03 19:29:48 EST 2017
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#19 by G. C. Greubel at Fri Mar 03 19:29:40 EST 2017
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| LINKS
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G. C. Greubel, <a href="/A124575/b124575.txt">Table of n, a(n) for the first 100 rows, flattened</a>
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| FORMULA
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T(n,k)=) = T(n-1,k-1)+4T) + 4*T(n-1,k)+) + T(n-1,k-1) for k>=2.
Sum_{k, k>=0} T(m,k)*T(n,k) = T(m+n,0)= A033543(m+n). [From _). - _Philippe Deléham_, Nov 22 2009]
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approved
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#18 by Bruno Berselli at Thu Jan 09 09:32:19 EST 2014
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#17 by Jean-François Alcover at Thu Jan 09 09:18:52 EST 2014
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#16 by Jean-François Alcover at Thu Jan 09 09:18:40 EST 2014
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| MATHEMATICA
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M[n_] := SparseArray[{{1, 1} -> 2, Band[{2, 2}] -> 4, Band[{1, 2}] -> 1, Band[{2, 1}] -> 1}, {n, n}]; row[1] = {1}; row[n_] := MatrixPower[M[n], n-1] // First // Normal; Table[row[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Jan 09 2014 *)
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approved
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#15 by N. J. A. Sloane at Sun Sep 08 19:59:15 EDT 2013
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| COMMENTS
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Column k=0 yields A033543 (2nd binomial transform of the sequence A000957(n+1)). Row sums yield A133158. [Corrected by _Philippe Deléham, _, Oct 24 2007, Dec 05 2009]
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Discussion
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| Sun Sep 08
| 19:59
| OEIS Server: https://oeis.org/edit/global/1941
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