Real-rooted atlas

A112007 → x ↦ x+1

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1314
21510126
3105105251236
494512604905612752
5103951732594501918119139208
61351352702701905755698068252461660032
7202702547297254099095163663530299522935501112818912
8344594259189180094594500475074601212211014872007431610121282137824
9654729075196418722523432409001422280860466876410814313506914908235092203516939897856
10137493105754583103525062199262125443469826801789286499041041600605108803703095092073173140821188666182784
113162341432251159525191825176449485712514467330127256947402962501991249360103330992665030496165701347796452252341817715617349020544
1279058535806253162341432250053338158823950494730748512002761014333577595404210902002026763158420254752658160175393368155751560366879678163681181790703209728
13213458046676875924984868933125171293494246875017740735434922501129764045234825458380554006375118885395048420192823952721401861763348445980494927232417578670209009223275116353726042486272
146190283353629375288879889836037505827404674278687566681131440423500477589101092853752229877374850125068597890721710351375295856374440174073797222325131315699451305361814583451004653127663259533655181238513970965257728
Technical details
Node id
n-fb19feff6ad7c4ad · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-014ae34d4ff839c2

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:         3,          1;
  2:        15,         10,          1;
  3:       105,        105,         25,          1;
  4:       945,       1260,        490,         56,         1;
  5:     10395,      17325,       9450,       1918,       119,        1;
  6:    135135,     270270,     190575,      56980,      6825,      246,       1;
  7:   2027025,    4729725,    4099095,    1636635,    302995,    22935,     501,      1;
  8:  34459425,   91891800,   94594500,   47507460,  12122110,  1487200,   74316,   1012,    1;
  9: 654729075, 1964187225, 2343240900, 1422280860, 466876410, 81431350, 6914908, 235092, 2035, 1;
%F Formula
The row polynomials satisfy P(n,x) = (2*n+1+n*x)*P(n-1,x) - (x+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 + n*x)*P[n - 1, x] - (x + 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 + n*x)*v[i - 1] - (x + 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