Real-rooted atlas

A104684 → basis A013610 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(\left(-3 + 3\,n\right)\,P_{n}(x) = \left(-78 - 45\,x + 48\,n + 27\,n x\right)\,P_{n-1}(x) + \left(-40 - 54\,x - 18\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2639
34642997
4396522216271161
53606622838349728114721
6338767245059400229504050243192969
7324556828108849150442800121500160387292582881
831518969345252115093447569450283500059194861236218735066313
930915046104454312150212916119343672567850501634320827147962274486561481003009
1030554355611588500981906051824177581905210259683563775505588720006411888532826686196836649718409
11303801987612780582300236647453502529960480017205955200772351297222940536504382748005019165029524505904992497952097
123035486685614027292298828875848760034843174245027272084400014460669115252611889176129943885502100176100205687350103926241771471293068265993
133045233439961533394121112347397250820446702012698004135592760750253059085699210910126636643313376383686956562690096781311008223163743613798853144118150727906689
143065412858696167056743152524130629832721661220562609492605159499087004198610535748220941857915264755860970779219582665513843554926947004316576378432194695781243571941594323255660741430857
1530946859111256181475744293368486296104985964787775207742432859906327336056667175687310132378079727271246158242282406208488565066029041098038404663217469974962441872542992321238044953642408991847829693611681892224097

Other recurrences

Image under the basis map A007318 · proved equal for all n (proof step)

\(\left(1 - n\right)\,P_{n}(x) = \left(-32 - 48\,x - 18\,x^{2} + 16\,n + 24\,n x + 9\,n x^{2}\right)\,P_{n-2}(x) + \left(18 + 9\,x - 12\,n - 6\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 6 + 3\,x\)
Technical details
Node id
n-7443809b677f8653 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-7fa92f96a89db90b)
Fingerprint
f6-42186a769c41ebed

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept -1)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:         1;
   2:         6,          3;
   3:        46,         42,          9;
   4:       396,        522,        216,         27;
   5:      3606,       6228,       3834,        972,         81;
   6:     33876,      72450,      59400,      22950,       4050,       243;
   7:    324556,     828108,     849150,     442800,     121500,     16038,      729;
   8:   3151896,    9345252,   11509344,    7569450,    2835000,    591948,    61236,     2187;
   9:  30915046,  104454312,  150212916,  119343672,   56785050,  16343208,  2714796,   227448,   6561;
  10: 305543556, 1158850098, 1906051824, 1775819052, 1025968356, 377550558, 87200064, 11888532, 826686, 19683;
%F Formula
The row polynomials satisfy (3*n-3)*P(n,x) = (48*n-78+(27*n-45)*x)*P(n-1,x) - (40+54*x+18*x^2)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((48*n - 78 + (27*n - 45)*x)*P[n - 1, x] - (40 + 54*x + 18*x^2)*D[P[n - 1, x], x])/(3*n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((48*n - 78 + (27*n - 45)*x)*v[i - 1] - (40 + 54*x + 18*x^2)*deriv(v[i - 1], x))/(3*n - 3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023