Real-rooted atlas

A142459 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 4\,x + 5\,x^{2} + 2\,x^{3}\right)\,P_{n-1}(x) + \left(4\,x + 12\,x^{2} + 12\,x^{3} + 4\,x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2123
31121225
4162180120363
513121992336016807345
61156219800636007560030240190803
7178121875721024800217560019958406652806056905
813906217350201519812051332400810734406054048017297280227215803
911953121584955221492576010854244802626525440329712768020756736005189184009834640225
101976562143817840295253040021461176800746946748801402117516801459198540807939451520017643225600482422523043
11148828121300219932398210736004061754312001954889354880514857663936078771813120006997614624000335221286400067044257280026448218974585
1212441406211731276260530676156120746247736080048363834214080171721981791360362301250671360467236721952000361950773184000154872234316800281585880576001602610293394443
1311220703121057279707127016102792160134291581319280114996511099008053570538974937601497150779604480026287849344556800292782271541760002010389098798080077717703038976001295295050649600106356973179941905
141610351562952284158280922666160052002381182321518000265665157945387201592255798791257605734571274201369601316644262251718400197708128241944320019368321623659008001194093085170585600420970891461120000647647525324800007672110061127082483
151305175781285742195338921208988849650400417718320898788006007581134726371204565102844457995840207701217720412416006065297447691081600011791791236607782400015453920738894077440013508113965204418560075555855599441817600244810764572774400003497296636753920000597704434705629473065
Technical details
Node id
n-8525597faaa26fd2 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-d6ac9abcd3bcb1be

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 -1)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1;
   2: 1,      2;
   3: 1,     12,        12;
   4: 1,     62,       180,        120;
   5: 1,    312,      1992,       3360,        1680;
   6: 1,   1562,     19800,      63600,       75600,       30240;
   7: 1,   7812,    187572,    1024800,     2175600,     1995840,       665280;
   8: 1,  39062,   1735020,   15198120,    51332400,    81073440,     60540480,     17297280;
   9: 1, 195312,  15849552,  214925760,  1085424480,  2626525440,   3297127680,   2075673600,   518918400;
  10: 1, 976562, 143817840, 2952530400, 21461176800, 74694674880, 140211751680, 145919854080, 79394515200, 17643225600;
%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) + (4*x+12*x^2+12*x^3+4*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((1 + 4*x + 5*x^2 + 2*x^3)*P[n - 1, x] + (4*x + 12*x^2 + 12*x^3 + 4*x^4)*D[P[n - 1, x], x])/(1 + 2*x + x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((1 + 4*x + 5*x^2 + 2*x^3)*v[i - 1] + (4*x + 12*x^2 + 12*x^3 + 4*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, v[i])))};
T(10)

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