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

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

Offset

0,5

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, May 31 2004

Diagonal sums are A052535. - Paul Barry, Jan 21 2005

Matrix inverse is the triangle of Salie numbers A098435. - Paul Barry, 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). - Clark Kimberling, Feb 13 2010

T(n,k) is the number of stack polyominoes of perimeter 2n+4 with k+1 columns. - Emanuele Munarini, Apr 07 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(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

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

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

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

Multiplying the n-th diagonal by A001147(n) generates A001497. - Tom Copeland, Oct 04 2016

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.

Molly Fenn and Eric Sommers, A transitivity result for ad-nilpotent ideals in type A, arXiv:2101.04091 [math.RT], 2021.

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, arXiv:cond-mat/0407444 See Eq. 2.27. Mentions this sequence. - N. J. A. Sloane, Mar 14 2014

Jay Kappraff and Gary W. Adamson, Polygons and Chaos, 5th Interdispl Symm. Congress and Exh. Jul 8-14, Sydney, 2001 - [with commercial pop-ups].

Jay Kappraff and Gary W. Adamson, Polygons and Chaos, Journal of Dynamical Systems and Geometric Theories, Vol. 2 pp. 79-94, (Nov 2004).

Formula

G.f.: (1-t*z)/((1-t*z)^2-z). - Emeric Deutsch, 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, 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 07 2011

T(n, k) = binomial(2*n-k, k) = A085478(n, n-k), for n >= 0, k = 0..n. - Wolfdieter Lang, Mar 25 2020

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; ...
...
(0, 1, 0, 0, 0, 0, ...) DELTA (1, 0, 1, 0, 0, 0, ...) begins:
  1;
  0, 1;
  0, 1, 1;
  0, 1, 3,  1;
  0, 1, 5,  6,  1;
  0, 1, 7, 15, 10,  1;
  0, 1, 9, 28, 35, 15, 1. Philippe Deléham, Apr 01 2012

Maple

T:=(n, k)->binomial(2*n-k, k): seq(seq(T(n, k), k=0..n), n=0..11);

Mathematica

Flatten[Table[Binomial[2n - k, k], {n, 0, 11}, {k, 0, n}]] (* Emanuele Munarini, Apr 07 2011 *)

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 07 2011 */
(Magma) [Binomial(2*n-k, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2019
(Sage) [[binomial(2*n-k, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Aug 01 2019
(GAP) Flat(List([0..12], n-> List([0..n], k-> Binomial(2*n-k, k) ))); # G. C. Greubel, Aug 01 2019

Crossrefs

These are the even-indexed rows of A011973, the odd-indexed rows form A053123.

Cf. A000108, A003558, A027989, A052535, A054142, A076756, A084938, A085478, A098435, A115139, A172431, A172991, A188648.

Cf. A001147, A001497.

Sequence in context: A239331 A145033 A202672 * A076756 A114172 A271942

Adjacent sequences: A054139 A054140 A054141 * A054143 A054144 A054145

Keyword

nonn,tabl

Author

Clark Kimberling

Status

approved