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Search: id:A055830
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| A055830 |
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Triangle T read by rows: diagonal differences of triangle A037027. |
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28
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| 1, 1, 0, 2, 1, 0, 3, 3, 1, 0, 5, 7, 4, 1, 0, 8, 15, 12, 5, 1, 0, 13, 30, 31, 18, 6, 1, 0, 21, 58, 73, 54, 25, 7, 1, 0, 34, 109, 162, 145, 85, 33, 8, 1, 0, 55, 201, 344, 361, 255, 125, 42, 9, 1, 0, 89, 365, 707, 850, 701, 413, 175, 52, 10, 1, 0, 144
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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Or, coefficients of a generalized Lucas-Pell polynomial read by rows. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 05 2006
Equals A046854(shifted) * Pascal's triangle; where A046854 is shifted down one row and "1" inserted at (0,0). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 24 2008]
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FORMULA
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G.f.: (1-yz) / [1-y(1+y+z)].
T(i, j) = R(i-j, j), where R(0, 0)=1, R(0, j)=0 for j >= 1, R(1, j)=1 for j >= 0, R(i, j)=SUM{R(i-2, k)+R(i-1, k): k=0, 1, ..., j} for i >= 1, j >= 1.
Sum_{k, 0<=k<=n}x^k*T(n,k)= A039834(n-2), A000012(n), A000045(n+1), A001333(n), A003688(n), A015448(n), A015449(n), A015451(n), A015453(n), A015454(n), A015455(n), A015456(n), A015457(n) for x= -2,-1,0,1,2,3,4,5,6,7,8,9,10 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 22 2006
Sum_{k, 0<=k<=[n/2]}T(n-k,k)=A011782(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 22 2006
Triangle T(n,k), 0<=k<=n, given by [1, 1, -1, 0, 0, 0, 0, 0, ...] DELTA [0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 05 2006
T(n,0)= Fibonacci(n+1)=A000045(n+1) . Sum_{k, 0<=k<=n}T(n,k)=A001333(n) . T(n,k)=0 if k>n or if k<0, T(0,0)=1, T(1,1)=0, T(n,k)=T(n-1,k-1)+T(n-1,k)+T(n-2,k) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 05 2006
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EXAMPLE
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1
1,0
2,1,0
3,3,1,0
5,7,4,1,0
8,15,12,5,1,0
13,30,31,18,6,1,0
21,58,73,54,25,7,1,0
34,109,162,145,85,33,8,1,0
55,201,344,361,255,125,42,9,1,0
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CROSSREFS
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Left-hand columns include A000045, A023610.
Right-hand columns include A055831, A055832, A055833, A055834, A055835, A055836, A055837, A055838, A055839, A055840.
Row sums: A001333 (numerators of continued fraction convergents to sqrt(2)).
Cf. A122075 (another version).
A046854 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 24 2008]
Sequence in context: A100224 A089000 A107238 this_sequence A079123 A121548 A113020
Adjacent sequences: A055827 A055828 A055829 this_sequence A055831 A055832 A055833
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KEYWORD
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nonn,tabl
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AUTHOR
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Clark Kimberling (ck6(AT)evansville.edu), May 28 2000
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EXTENSIONS
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Edited by Ralf Stephan, Jan 12 2005
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