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A054142 Triangular array binomial(2*n-k, k), k=0..n, n >= 0.

%I A054142 #93 Feb 05 2022 16:42:25
%S A054142 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,
%T A054142 13,66,165,210,126,28,1,1,15,91,286,495,462,210,36,1,1,17,120,455,
%U A054142 1001,1287,924,330,45,1,1,19,153,680,1820,3003,3003,1716,495,55,1
%N A054142 Triangular array binomial(2*n-k, k), k=0..n, n >= 0.
%C A054142 Row sums are odd-indexed Fibonacci numbers.
%C A054142 T(n,k) is the number of nondecreasing Dyck paths of semilength n+1, having k double rises. Mirror image of A085478. - _Emeric Deutsch_, May 31 2004
%C A054142 Diagonal sums are A052535. - _Paul Barry_, Jan 21 2005
%C A054142 Matrix inverse is the triangle of Salie numbers A098435. - _Paul Barry_, Jan 21 2005
%C A054142 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). - _Clark Kimberling_, Feb 13 2010
%C A054142 T(n,k) is the number of stack polyominoes of perimeter 2n+4 with k+1 columns. - _Emanuele Munarini_, Apr 07 2011
%C A054142 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(2*Pi/N) = 3.24697... = A116415; we obtain the trajectory (3.24697...-> 1.55495...-> 0.198062...; 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(2*Pi/N). Check: given 2*cos(2*Pi/7) = 1.24697..., we obtain the 3-cycle using (x^2 - 2): (1.24697...-> -0.445041...-> 1.801937...; where the terms in either set are intermediate terms in the other, irrespective of sign. - _Gary W. Adamson_, Sep 22 2011
%C A054142 A054142 is jointly generated with A172431 as an array of coefficients of polynomials u(n,x): initially, u(1,x)=v(1,x)=1; for n>1, u(n,x)=x*u(n-1,x)+v(n-1,x) and v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x). See the Mathematica section of A172431. - _Clark Kimberling_, Mar 09 2012
%C A054142 Subtriangle of the triangle given by (0, 1, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Apr 01 2012
%C A054142 The o.g.f. for row n of the array A(n, k) = binomial(2*n-k,k), k >= 0, n >= 0 is G(n,x) = Sum_{k=0..n} T(n, k)*x^k + (-x)^(2*n+1) * c(-x)^(2*n+1) / sqrt(1-4*(-x)), for n >= 0. Here c(x) is the o.g.f. of A000108 (Catalan). For powers of c(x) see the W. Lang link in A115139. For the alternating sign case replace x by -x. - _Wolfdieter Lang_, Sep 12 2016
%C A054142 Multiplying the n-th diagonal by A001147(n) generates A001497. - _Tom Copeland_, Oct 04 2016
%H A054142 Vincenzo Librandi, <a href="/A054142/b054142.txt">Table of n, a(n) for n = 0..495</a>
%H A054142 E. Barcucci, A. Del Lungo, S. Fezzi and R. Pinzani, <a href="http://dx.doi.org/10.1016/S0012-365X(97)82778-1">Nondecreasing Dyck paths and q-Fibonacci numbers</a>, Discrete Math., 170, 1997, 211-217.
%H A054142 D. Dumont and J. Zeng, <a href="http://dx.doi.org/10.1023/A:1009759202242">Polynomes d'Euler et les fractions continues de Stieltjes-Rogers</a>, Ramanujan J. 2 (1998) 3, 387-410.
%H A054142 Molly Fenn and Eric Sommers, <a href="https://arxiv.org/abs/2101.04091">A transitivity result for ad-nilpotent ideals in type A</a>, arXiv:2101.04091 [math.RT], 2021.
%H A054142 J. L. Jacobsen, and J. Salas, Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models IV. Chromatic polynomial with cyclic boundary conditions, J. Stat. Phys. 122 (2006) 705-760, <a href="http://arxiv.org/abs/cond-mat/0407444">arXiv:cond-mat/0407444</a> See Eq. 2.27. Mentions this sequence. - _N. J. A. Sloane_, Mar 14 2014
%H A054142 Jay Kappraff and Gary W. Adamson, <a href="http://vismath7.tripod.com/proceedings/kappraff.htm">Polygons and Chaos</a>, 5th Interdispl Symm. Congress and Exh. Jul 8-14, Sydney, 2001 - [with commercial pop-ups].
%H A054142 Jay Kappraff and Gary W. Adamson, <a href="http://archive.bridgesmathart.org/2001/bridges2001-67.pdf">Polygons and Chaos</a>, Journal of Dynamical Systems and Geometric Theories, Vol. 2 pp. 79-94, (Nov 2004).
%F A054142 G.f.: (1-t*z)/((1-t*z)^2-z). - _Emeric Deutsch_, May 31 2004
%F A054142 Column k has g.f.: (Sum_{j=0..k+1} binomial(k+1, 2j)*x^j)*x^k/(1-x)^(k+1). - _Paul Barry_, Jun 22 2005
%F A054142 Recurrence: T(n+2,k+2) = T(n+1,k+2) + 2*T(n+1,k+1) - T(n,k). - _Emanuele Munarini_, Apr 07 2011
%F A054142 T(n, k) = binomial(2*n-k, k) = A085478(n, n-k), for n >= 0, k = 0..n. - _Wolfdieter Lang_, Mar 25 2020
%e A054142 Triangle begins:
%e A054142   1;
%e A054142   1,  1;
%e A054142   1,  3,  1;
%e A054142   1,  5,  6,   1;
%e A054142   1,  7, 15,  10,   1;
%e A054142   1,  9, 28,  35,  15,   1;
%e A054142   1, 11, 45,  84,  70,  21,   1;
%e A054142   1, 13, 66, 165, 210, 126,  28,  1;
%e A054142   1, 15, 91, 286, 495, 462, 210, 36, 1; ...
%e A054142 ...
%e A054142 (0, 1, 0, 0, 0, 0, ...) DELTA (1, 0, 1, 0, 0, 0, ...) begins:
%e A054142   1;
%e A054142   0, 1;
%e A054142   0, 1, 1;
%e A054142   0, 1, 3,  1;
%e A054142   0, 1, 5,  6,  1;
%e A054142   0, 1, 7, 15, 10,  1;
%e A054142   0, 1, 9, 28, 35, 15, 1. _Philippe Deléham_, Apr 01 2012
%p A054142 T:=(n,k)->binomial(2*n-k,k): seq(seq(T(n,k), k=0..n), n=0..11);
%t A054142 Flatten[Table[Binomial[2n - k, k], {n, 0, 11}, {k, 0, n}]] (* _Emanuele Munarini_, Apr 07 2011 *)
%o A054142 (PARI) T(n,k)=if(n<0,0,polcoeff(charpoly(matrix(n,n,i,j,-min(i,j))),k))
%o A054142 (Maxima) create_list(binomial(2*n-k,k),n,0,10,k,0,n); /* _Emanuele Munarini_, Apr 07 2011 */
%o A054142 (Magma) [Binomial(2*n-k,k): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Aug 01 2019
%o A054142 (Sage) [[binomial(2*n-k,k) for k in (0..n)] for n in (0..12)] # _G. C. Greubel_, Aug 01 2019
%o A054142 (GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(2*n-k,k) ))); # _G. C. Greubel_, Aug 01 2019
%Y A054142 These are the even-indexed rows of A011973, the odd-indexed rows form A053123.
%Y A054142 Cf. A000108, A003558, A027989, A052535, A054142, A076756, A084938, A085478, A098435, A115139, A172431, A172991, A188648.
%Y A054142 Cf. A001147, A001497.
%K A054142 nonn,tabl
%O A054142 0,5
%A A054142 _Clark Kimberling_