Real-rooted atlas

A185410 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1112
23418
3152713156
4105240174401560
59452625255099012117232
61039534020414752280053133641114368
713513550935574749553287518616527657109312139776
8202702586486401484406013018320609903014414401415643280146223360
9344594251641890253227023803361743001980927906498135010797900717660984111132124672
106547290753445942500763174912592108016006549286590276385284066041745079173600361782329524130999600128
11137493105757922221807519521264262526790175912522330274967011554310691036346138290649797225057230431518177687885731938366468096
1231623414322519799007228005372706789450826337011500079038711501754851601634880190186791102045828927144062487880455409701864091154154265720131114518056960
13790585358062553443570205025158364559202850269830874763450291369841737975207139834076175974983073664602976546482910056035729099355910700545752913490125045656793079716111121542540992512
142134580466768751549547301802500497847411679117593088332316980001120247536305397590574041723165004983851789791875184942565327520044859034478758566958932500460552357479992520622714840022852311332391484143664751042265088
Technical details
Node id
n-76ffafa4c38c7321 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-1afcaeeccd1ccf36

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:        1,         1;
  2:        3,         4,         1;
  3:       15,        27,        13,         1;
  4:      105,       240,       174,        40,         1;
  5:      945,      2625,      2550,       990,       121,        1;
  6:    10395,     34020,     41475,     22800,      5313,      364,        1;
  7:   135135,    509355,    747495,    532875,    186165,    27657,     1093,      1;
  8:  2027025,   8648640,  14844060,  13018320,   6099030,  1441440,   141564,   3280,    1;
  9: 34459425, 164189025, 322702380, 336174300, 198092790, 64981350, 10797900, 717660, 9841, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2*n-1+(2*n-1)*x)*P(n-1,x) - (2*x+2*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[(2*n - 1 + (2*n - 1)*x)*P[n - 1, x] - (2*x + 2*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; for(n = 1, N, my(i = n + 1); v[i] = (2*n - 1 + (2*n - 1)*x)*v[i - 1] - (2*x + 2*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

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