Real-rooted atlas

A114687 → basis A013609 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(2 + 19\,n + 12\,n x\right)\,P_{n-1}(x) + \left(-15 - 22\,x - 8\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
17411
2556016131
3469744384641661
4423787766656217625622101
53990710182010160049280115201024305151
6387739117454814517609328003264005836840964335711
73858505135268001993331215944320743680020070402867201638463009881

All 15 rows

Other recurrences

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

\(\left(2 + n\right)\,P_{n}(x) = \left(25 + 40\,x + 16\,x^{2} - 25\,n - 40\,n x - 16\,n x^{2}\right)\,P_{n-2}(x) + \left(7 + 4\,x + 14\,n + 8\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 7 + 4\,x\)
Technical details
Node id
n-3177bea1e7729b42 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-bc62b354048e24a0)
Fingerprint
f6-b0d7ee3d6c150f9e

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:         1;
  1:         7,          4;
  2:        55,         60,         16;
  3:       469,        744,        384,         64;
  4:      4237,       8776,       6656,       2176,        256;
  5:     39907,     101820,     101600,      49280,      11520,       1024;
  6:    387739,    1174548,    1451760,     932800,     326400,      58368,      4096;
  7:   3858505,   13526800,   19933312,   15944320,    7436800,    2007040,    286720,    16384;
  8:  39130777,  155795472,  266552832,  255155712,  148829184,   53846016,  11698176,  1376256,   65536;
  9: 402972031, 1795877748, 3498441408, 3899524608, 2731373568, 1240952832, 363331584, 65470464, 6488064, 262144;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = (19*n+2+12*n*x)*P(n-1,x) - (15+22*x+8*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((19*n + 2 + 12*n*x)*P[n - 1, x] - (15 + 22*x + 8*x^2)*D[P[n - 1, x], x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((19*n + 2 + 12*n*x)*v[i - 1] - (15 + 22*x + 8*x^2)*deriv(v[i - 1], x))/(n + 2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

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