Real-rooted atlas

A142458 → x ↦ x/(x+1)

nonnegative coefficients

Proved for all n all roots real, in [−1, 0]coefficients of one signHadamard kernel

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 4\,x + 5\,x^{2} + 2\,x^{3}\right)\,P_{n-1}(x) + \left(3\,x + 9\,x^{2} + 9\,x^{3} + 3\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
21101021
314212080243
41170105017608803861
51682820026000308001232078003
612730608103256006864006283202094401913301
71109224393203742480125048001966272014660800418880055209843
8143690312985040939360203729680489670720612821440385369600963424001832046741
9117476222127400434432400309881880010686947040199680096002073456000011272060800250490240068722033203
101699050155765610452134320045063400800214374615840561070282080855520512000758696400000363210848000726421696002875256036181
1112796202109385452046459556880635558985600406088146464014304703828800300428569056003864438177600029896010144000127850218496002324549427200132741520589043
121111848107670962650473346404960877332193848073871929232640340824357646080947036928499200165709525322880018418422102080001263319971513600488155379712000813592299520006702759610483221
13144739242537526626004794831750800119259995654800130477737486096077314855922304002765129957990400063209230686201600946560584697728009257634321295360057023521998643200200957297981440003091650738176000367464206075694003
141178956970376492334410483783388088001603123092771200225460779369425601690794416250139207629583026024960002216210657492832000429380963499488000056148923352249728004901242730768384000273936527248384000088730376185651200012675768026521600021735817773353960661
Technical details
Node id
n-7e2941a6c64b355d · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-72ea6b42576fa2d7

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,      2;
  2: 1,     10,       10;
  3: 1,     42,      120,        80;
  4: 1,    170,     1050,      1760,        880;
  5: 1,    682,     8200,     26000,      30800,       12320;
  6: 1,   2730,    60810,    325600,     686400,      628320,      209440;
  7: 1,  10922,   439320,   3742480,   12504800,    19662720,    14660800,     4188800;
  8: 1,  43690,  3129850,  40939360,  203729680,   489670720,   612821440,   385369600,    96342400;
  9: 1, 174762, 22127400, 434432400, 3098818800, 10686947040, 19968009600, 20734560000, 11272060800, 2504902400;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (1+4*x+5*x^2+2*x^3)*P(n-1,x) + (3*x+9*x^2+9*x^3+3*x^4)*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[((1 + 4*x + 5*x^2 + 2*x^3)*P[n - 1, x] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
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] = ((1 + 4*x + 5*x^2 + 2*x^3)*v[i - 1] + (3*x + 9*x^2 + 9*x^3 + 3*x^4)*deriv(v[i - 1], x))/(1 + 2*x + x^2)); 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