Real-rooted atlas

A008459 → basis A013611 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(n\right)\,P_{n}(x) = \left(-10 - 8\,x + 16\,n + 12\,n x\right)\,P_{n-1}(x) + \left(-10 - 18\,x - 8\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
16410
2465616118
3396696384641540
4360683046816230425621286
5338769660010560054400128001024304300
63245561104144150960010496003840006758440964443580
73151896124603362046105617942400896000024944643440641638465830600
8309150461392724162670451842828887041794688006887014415253504170393665536985483270
93055435561545133464338853657642093488643242566656159099494448994713689063424825753626214414869654300
1030380198761704077640042070658400599694336005437931520032546820096128894976003283353600501350400393216001048576225759595348
11303548668561870305639845133484224008259122784008619325440006093714063362956080906249734799360020978073600273940480018454937641943043444812388280
123045233439962044525494816617595112569611070106713600130705153920001066388904345661300245135362482229477376694871654400128896204800146024693768556380161677721652781007848284
13306541285869622274232420336734334192483841451154076669441912602861312001769291024117761176650891919365662563395174419560578678784473455656960076652583321676227280896392586854467108864811510465220920
1430946859111256241967659057824864526408863936186731901094502427177286394818562811472855166976212429981193830411854785344962564880140248023041462399936757763102270396825644336142090243908420239361785095782426843545612513859077134008

Other recurrences

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

\(\left(n\right)\,P_{n}(x) = \left(16 + 32\,x + 16\,x^{2} - 16\,n - 32\,n x - 16\,n x^{2}\right)\,P_{n-2}(x) + \left(-6 - 4\,x + 12\,n + 8\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 4\,x\)
Technical details
Node id
n-f73887acc725d564 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
derived:basis_map_A007318 (equal by e-ae8ca04dd20b1f4f)
Fingerprint
f6-48da6b31a66f55a0

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:         6,          4;
  2:        46,         56,         16;
  3:       396,        696,        384,         64;
  4:      3606,       8304,       6816,       2304,        256;
  5:     33876,      96600,     105600,      54400,      12800,       1024;
  6:    324556,    1104144,    1509600,    1049600,     384000,      67584,      4096;
  7:   3151896,   12460336,   20461056,   17942400,    8960000,    2494464,    344064,    16384;
  8:  30915046,  139272416,  267045184,  282888704,  179468800,   68870144,  15253504,  1703936,   65536;
  9: 305543556, 1545133464, 3388536576, 4209348864, 3242566656, 1590994944, 489947136, 89063424, 8257536, 262144;
%F Formula
The row polynomials satisfy n*P(n,x) = (16*n-10+(12*n-8)*x)*P(n-1,x) - (10+18*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[((16*n - 10 + (12*n - 8)*x)*P[n - 1, x] - (10 + 18*x + 8*x^2)*D[P[n - 1, x], x])/(n)]];
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] = ((16*n - 10 + (12*n - 8)*x)*v[i - 1] - (10 + 18*x + 8*x^2)*deriv(v[i - 1], x))/(n)); 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