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A054142
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Triangular array C(2n-k,k), k=0,1,...,n, n >= 0.
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25
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1, 1, 1, 1, 3, 1, 1, 5, 6, 1, 1, 7, 15, 10, 1, 1, 9, 28, 35, 15, 1, 1, 11, 45, 84, 70, 21, 1, 1, 13, 66, 165, 210, 126, 28, 1, 1, 15, 91, 286, 495, 462, 210, 36, 1, 1, 17, 120, 455, 1001, 1287, 924, 330, 45, 1, 1, 19, 153, 680, 1820, 3003, 3003, 1716, 495, 55, 1, 1, 21, 190
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Row sums are odd-indexed Fibonacci numbers.
T(n,k) is the number of nondecreasing Dyck paths of semilength n+1, having k double rises. Mirror image of A085478. - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 31 2004
Diagonal sums are A052535. - Paul Barry (pbarry(AT)wit.ie), Jan 21 2005
Matrix inverse is the triangle of Salie numbers A098435. - Paul Barry (pbarry(AT)wit.ie), Jan 21 2005
Coefficients of Morgan-Voyce polynomial b(n,x); e.g., b(3,x)=x^3+5x^2+6x+1. See A172431 for coefficients of Morgan-Voyce polynomial B(n,x).
T(n,k) is the number of stack polyominoes of perimeter 2n+4 with k+1 columns. [Emanuele Munarini, Apr 7 2011]
Roots of signed n-th polynomials are chaotic with respect to the operation (-2, x^2), with cycle lengths A003558(n). Example: starting with a root to x^3 - 5x^2 + 6x - 1 = 0; (2 + 2*Cos (2Pi/N) = 3.24697... = A116415; we obtain the trajectory (3.24697...-> 1.55495...-> .198062...; the 3 roots to the polynomial with cycle length 3 matching A003558(3) = 3. The operation (-2, x^2) is the reversal of the well known chaotic operation (x^2 - 2) [Kappraff, Adamson, 2004] starting with seed 2*Cos (2Pi/N). Check: given 2*Cos (2Pi/7) = 1.24697..., we obtain the 3-cycle using (x^2 - 2): (1.24697...-> -.445041...-> 1.801937...; where the terms in either set are intermediate terms in the other, irrespective of sign. - Gary W. Adamson, Sep 22 2011
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REFERENCES
| Jay Kappraff and Gary W. Adamson, Polygons and Chaos, Journal of Dynamical Systems and Geometric Theories, Vol. 2 pp. 79-94, (Nov 2004)
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..495
E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, Nondecreasing Dyck paths and q-Fibonacci numbers, Discrete Math., 170, 1997, 211-217.
D. Dumont and J. Zeng, Polynomes d'Euler et les fractions continues de Stieltjes-Rogers, Ramanujan J. 2 (1998) 3, 387-410.
Jay Kappraff and Gary W. Adamson, Polygons and Chaos, 5th Interdispl Symm. Congress and Exh. Jul 8 -14, Sydney, 2001 - [with commercial pop-ups].
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FORMULA
| G.f.: (1-t*z)/((1-t*z)^2-z). - Emeric Deutsch (deutsch(AT)duke.poly.edu), May 31 2004
Column k has g.f. sum{j=0..k+1, binomial(k+1, 2j)x^j}*x^k/(1-x)^(k+1); - Paul Barry (pbarry(AT)wit.ie), Jun 22 2005
Recurrence: T(n+2,k+2) = T(n+1,k+2) + 2*T(n+1,k+1) - T(n,k) [Emanuele Munarini, Apr 7 2011]
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EXAMPLE
| Triangle begins:
1;
1, 1;
1, 3, 1;
1, 5, 6, 1;
1, 7, 15, 10, 1;
1, 9, 28, 35, 15, 1;
1, 11, 45, 84, 70, 21, 1;
1, 13, 66, 165, 210, 126, 28, 1;
1, 15, 91, 286, 495, 462, 210, 36, 1; ...
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MAPLE
| T:=(n, k)->binomial(2*n-k, k): seq(seq(T(n, k), k=0..n), n=0..11);
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MATHEMATICA
| Flatten[Table[Binomial[2n - k, k], {n, 0, 11}, {k, 0, n}]] [Emanuele Munarini, Apr 7 2011]
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PROG
| (PARI) T(n, k)=if(n<0, 0, polcoeff(charpoly(matrix(n, n, i, j, -min(i, j))), k))
(Maxima) create_list(binomial(2*n-k, k), n, 0, 10, k, 0, n); [Emanuele Munarini, Apr 7 2011]
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CROSSREFS
| These are the even-indexed rows of A011973, the odd-indexed rows form A053123.
Cf. A076756, A085478, A172991, A188648, A027989.
Cf. A003558
Sequence in context: A145033 A202672 A076756 * A114172 A121522 A204027
Adjacent sequences: A054139 A054140 A054141 * A054143 A054144 A054145
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KEYWORD
| nonn,tabl
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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EXTENSIONS
| Added a comment Clark Kimberling (ck6(AT)evansville.edu), Feb 13 2010
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