Real-rooted atlas

A144441 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(4 + 4\,x\right)\,P_{n-2}(x) + \left(-2 - 3\,x + 4\,n + 4\,n x\right)\,P_{n-1}(x) + \left(-4\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
21616133
3168252861507
4241648322852436110537
544160110400926882863221861278067
69811842943552320516815044162725521093618917809
7256874248990598412031996876034960224062002527332546861336936555
8774547456309818982449192837763914186944161338899231768787223117852273436114660676153
92643736320011896813440021858978688020987962688011163269232031974977200437144892820992003213671861722065317027
10100771799142450385899571201051571193600011831308001280772502020678429275604357766061381202885902160153618985771526835936139712973663297
11424299050926082333644780093445453122659225607036716115814405452445066903042581062654347527253464102099211148975942240787111740360171336787323417968612412616929293019
121955806506225664117348390373539843035202691101286444190776712652800396567236957288962256546391789260880625535325548801726569667288448200795836097872104114110643521544355304521708984361160456121834301257
139796004493165363263674029205574860818059675939191971842928678553942349824298838300849602304019888492025843320328653474090911354882398831054620707843992003241354950435625059502411201369613705747681391081883432854492186111595430102153251603
14529766565233419878437083659566339391488114443131094808219648204571212206437228544233969765036118745088178439932952674627584917149580278188116483137119209845614284868830278116634039689032906167775286406252232372852500817950270620218561252554749775242724609361904742164944038803793
Technical details
Node id
n-6e03aa63410fab4d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-bb41b383faa3577c

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:           2,            1;
  2:          16,           16,            1;
  3:         168,          252,           86,            1;
  4:        2416,         4832,         2852,          436,            1;
  5:       44160,       110400,        92688,        28632,         2186,           1;
  6:      981184,      2943552,      3205168,      1504416,       272552,       10936,          1;
  7:    25687424,     89905984,    120319968,     76034960,     22406200,     2527332,      54686,         1;
  8:   774547456,   3098189824,   4919283776,   3914186944,   1613388992,   317687872,   23117852,    273436,       1;
  9: 26437363200, 118968134400, 218589786880, 209879626880, 111632692320, 31974977200, 4371448928, 209920032, 1367186, 1;
%F Formula
The row polynomials satisfy P(n,x) = (4+4*x)*P(n-2,x) + (4*n-2+(4*n-3)*x)*P(n-1,x) - (4*x+4*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[n_, x_] := P[n, x] = Expand[(4 + 4*x)*P[n - 2, x] + (4*n - 2 + (4*n - 3)*x)*P[n - 1, x] - (4*x + 4*x^2)*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] = 1; v[2] = 2 + x; for(n = 2, N, my(i = n + 1); v[i] = (4 + 4*x)*v[i - 2] + (4*n - 2 + (4*n - 3)*x)*v[i - 1] - (4*x + 4*x^2)*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

Same fingerprint ?

References

  1. [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