Real-rooted atlas

A075501 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(7 + x\right)\,P_{n-1}(x) + \left(6 + 6\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 55 + 20\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16Pn(1)
05520176
1505327391872
254975224108664111872
36982386441270102710951186080
41007407151767664841197040568513213276352
5161579052851812715584205321085127723510605175163759104
6284214097573904016382413340102509120121232866759201817222411355858176
75432922775123339263859684921636323610020449860858637786518146672429196279131225762304
81120340177352819930254122545972411411025540328001982656191019495118161020818822910720445953401773059428352
92476196276617683311024178369653131974913294285546981775662806490956334468546461282982246847722538180565395407120446915094528
1058332035387017174890387789688198471200839770107938359227776302088359537914546706527440383836219404188151480005463783279395320927304801574790732959744
111457666574447247471356148673020158887336698452863621942244425154118081346961849721225508136761521841900177812133200463110049373303265112505664715639910127842559117098196945371136
123848503494151193513339876994008329218179446161344340712477681305131806446594812243627011982055988662148412074009835197318948004565843241082893696131189401533322181368189250126241720816441536171562141859840
131069787864231083297395420258352391561558334761755192331774419490281743443271186383278920239992545342196345533223565496964095624061578912111052432332185872926707511494520814205266742655179401967957338657255226905735117665831366932856832
143121373055076107676912247613555627278441619434107498977132485616105272015996291315275800965650677807996209983361444921588163512934506209188424364613117516135488241120553441805181039029058244320296315775630165629625863687091301308580059202938808321609785691961755566080
Technical details
Node id
n-679a382cbe44b97b · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-388ff5e3b0d53753

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:            55,            20,             1;
  1:           505,           327,            39,             1;
  2:          5497,          5224,          1086,            64,            1;
  3:         69823,         86441,         27010,          2710,           95,           1;
  4:       1007407,       1517676,        648411,         97040,         5685,         132,          1;
  5:      16157905,      28518127,      15584205,       3210851,       277235,       10605,        175,         1;
  6:     284214097,     573904016,     382413340,     102509120,     12123286,      675920,      18172,       224,       1;
  7:    5432922775,   12333926385,    9684921636,    3236100204,    498608586,    37786518,    1466724,     29196,     279,     1;
  8:  112034017735,  281993025412,  254597241141,  102554032800,  19826561910,  1949511816,  102081882,   2910720,   44595,   340,   1;
  9: 2476196276617, 6833110241783, 6965313197491, 3294285546981, 775662806490, 95633446854, 6461282982, 246847722, 5381805, 65395, 407, 1;
%F Formula
The row polynomials satisfy P(n,x) = (7+x)*P(n-1,x) + (6+6*x)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 55 + 20*x + x^2.
%t Mathematica
P[0, x_] := 55 + 20*x + x^2;
P[n_, x_] := P[n, x] = Expand[(7 + x)*P[n - 1, x] + (6 + 6*x)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 55 + 20*x + x^2; for(n = 1, N, my(i = n + 1); v[i] = (7 + x)*v[i - 1] + (6 + 6*x)*deriv(v[i - 1], x)); 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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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.