Real-rooted atlas

A144436 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(4\,x + 12\,x^{2} + 12\,x^{3} + 4\,x^{4}\right)\,P_{n-2}(x) + \left(1 + 4\,x + 5\,x^{2} + 2\,x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
21101021
31267248147
41583385602801237
51122129639844680187211955
61250449022704469124267214224131253
71506147121144483692005721604244801212801616787
811018466585356002527016592662473348164593664114841622113813
9120421450562389904158297205265643296595520994144005386176011969280332864115
101409044537010330544934271524231673761078578224162112512014271993606812787201362557445471811701
11181861356552436920485289283203172445664107861638082219463456028218960000216978854409256203264168294604897583223891
12116378411057818199224029062054962262327468899776644416269853369344464521164416511361752320348928888064134539929600224233216001877120669141
13132762124136167498702241562118308015546736257687160492966430128745479686741293997312996048354560096628644259845925442166784208427886284832065828659238751351625011
1416553037404650306564318482599677792103888406488072916239664163156543549747289266036962560169955273564160219760359354368190500841674752106060248655872342969730969604899567585280854720947213877
Technical details
Node id
n-1271c9653b4e123c · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-1c1d701204aa53d8

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,   26,     72,      48;
  4: 1,   58,    338,     560,      280;
  5: 1,  122,   1296,    3984,     4680,     1872;
  6: 1,  250,   4490,   22704,    46912,    42672,    14224;
  7: 1,  506,  14712,  114448,   369200,   572160,   424480,   121280;
  8: 1, 1018,  46658,  535600,  2527016,  5926624,  7334816,  4593664,  1148416;
  9: 1, 2042, 145056, 2389904, 15829720, 52656432, 96595520, 99414400, 53861760, 11969280;
%F Formula
The row polynomials satisfy (1+2*x+x^2)*P(n,x) = (4*x+12*x^2+12*x^3+4*x^4)*P(n-2,x) + (1+4*x+5*x^2+2*x^3)*P(n-1,x) + (x+3*x^2+3*x^3+x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*x + 12*x^2 + 12*x^3 + 4*x^4)*P[n - 2, x] + (1 + 4*x + 5*x^2 + 2*x^3)*P[n - 1, x] + (x + 3*x^2 + 3*x^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; v[2] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = ((4*x + 12*x^2 + 12*x^3 + 4*x^4)*v[i - 2] + (1 + 4*x + 5*x^2 + 2*x^3)*v[i - 1] + (x + 3*x^2 + 3*x^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