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Search: id:A089942
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| 1, 0, 1, 1, 1, 1, 1, 3, 2, 1, 3, 6, 6, 3, 1, 6, 15, 15, 10, 4, 1, 15, 36, 40, 29, 15, 5, 1, 36, 91, 105, 84, 49, 21, 6, 1, 91, 232, 280, 238, 154, 76, 28, 7, 1, 232, 603, 750, 672, 468, 258, 111, 36, 8, 1, 603, 1585, 2025, 1890, 1398, 837, 405, 155, 45, 9, 1, 1585, 4213, 5500
(list; table; graph; listen)
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OFFSET
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0,8
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COMMENT
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Reverse of A071947 - related to lattice paths. First column is A005043.
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)=T(n-1,1), T(n,k)=T(n-1,k-1)+T(n-1,k)+T(n-1,k+1)for k>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Feb 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
Riordan array (f(x),x*g(x)), where f(x)is the o.g.f. of A005043 and g(x)is the o.g.f. of A001006. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 22 2009]
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REFERENCES
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D. Merlini, D. G. Rogers, R. Sprugnoli and M. C. Verri, On some alternative characterizations of Riordan arrays, Canad J. Math., 49 (1997), 301-320.
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FORMULA
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G.f.=(1+z-q)/[(1+z)(2z-t+tz+tq)], where q = sqrt(1-2z-3z^2).
Sum_{k, k>=0}T(m,k)*T(n,k)=T(m+n,0)=A005043(m+n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 22 2007
Sum_{k, 0<=k<=n}T(n,k)*(2k+1)=3^n . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 22 2007
Sum_{k, 0<=k<=n}T(n,k)*2^k = A112657(n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Apr 01 2007
T(n,2k)+T(n,2k+1)=A109195(n,k). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 11 2008]
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EXAMPLE
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1; 0,1; 1,1,1; 1,3,2,1; 3,6,6,3,1; 6,15,15,10,4,1
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CROSSREFS
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Row sums give A002426 (central trinomial coefficients).
Sequence in context: A115215 A158275 A147750 this_sequence A097409 A078268 A124782
Adjacent sequences: A089939 A089940 A089941 this_sequence A089943 A089944 A089945
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KEYWORD
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nonn,tabl,new
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AUTHOR
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Paul Barry (pbarry(AT)wit.ie), Nov 16 2003
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EXTENSIONS
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Edited by Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 04 2004
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