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A089942
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Inverse binomial matrix applied to A039599.
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30
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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; refs; listen; history; internal format)
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OFFSET
| 0,8
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COMMENTS
| 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]
Riordan array ((1+x-sqrt(1-2x-3x^2))/(2x(1+x)), (1-x-sqrt(1-2x-3x^2))/(2x)). Inverse of Riordan array
((1+x)/(1+x+x^2),x/(1+x+x^2)). E.g.f. of column k is exp(x)*(Bessel_I(k,2x)-Bessel_I(k+1,2x)).
Diagonal sums are A187306.
Simultaneous equations using the first n rows solve for diagonal lengths of odd N = (2n+1) regular polygons, with constants c^0, c^1, c^2,...; where c = (1 + 2*Cos 2Pi/N) = (Sin 3*Pi/N)/(Sin Pi/N) = the third longest diagonal of N>5. By way of example, take the first 4 rows relating to the nonagon, N=(2*4 + 1), with c = (1 + 2*Cos 2Pi/9) = 2.5320888.... The simultaneous equations are (1,0,0,0) = 1; (0,1,0,0) = c; (1,1,1,0) = c^2, (1,3,2,1) = c^3. The answers are 1, 2.532..., 2.879..., and 1.879...; the four distinct diagonal lengths of the nonagon with edge = 1. - Gary W. Adamson, Sep 07 2011
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REFERENCES
| 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.
E. Deutsch, L. Ferrari and S. Rinaldi, Production Matrices, Advances in Mathematics, 34 (2005) pp. 101-122.
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FORMULA
| 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
| Triangle begins
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
Production matrix is
0, 1,
1, 1, 1,
0, 1, 1, 1,
0, 0, 1, 1, 1,
0, 0, 0, 1, 1, 1,
0, 0, 0, 0, 1, 1, 1,
0, 0, 0, 0, 0, 1, 1, 1,
0, 0, 0, 0, 0, 0, 1, 1, 1,
0, 0, 0, 0, 0, 0, 0, 1, 1, 1
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CROSSREFS
| Row sums give A002426 (central trinomial coefficients).
Sequence in context: A115215 A158275 A147750 * A097409 A078268 A124782
Adjacent sequences: A089939 A089940 A089941 * A089943 A089944 A089945
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KEYWORD
| nonn,tabl
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Nov 16 2003
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EXTENSIONS
| Edited by Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 04 2004
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