Real-rooted atlas

A166344 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(1 + 2\,x + x^{2}\right)\,P_{n}(x) = \left(1 + 4\,x + 5\,x^{2} + 2\,x^{3}\right)\,P_{n-1}(x) + \left(x + 3\,x^{2} + 3\,x^{3} + x^{4}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1 + 8\,x + 8\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
118817
2118483299
3138198320160717
4178708207224009606219
511582358111202236020160672062877
6131875485391216740025512018816053760726219
716382359824584011065602535120310296019353604838409433917
8112787270810777526762000218500803946656040306560217728004838400136148139

All 15 rows

Technical details
Node id
n-943c738f9048f5dc · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ea33b747599ac881

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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,    8,      8;
   2: 1,   18,     48,       32;
   3: 1,   38,    198,      320,       160;
   4: 1,   78,    708,     2072,      2400,        960;
   5: 1,  158,   2358,    11120,     22360,      20160,       6720;
   6: 1,  318,   7548,    53912,    167400,     255120,     188160,      53760;
   7: 1,  638,  23598,   245840,   1106560,    2535120,    3102960,    1935360,      483840;
   8: 1, 1278,  72708,  1077752,   6762000,   21850080,   39466560,   40306560,    21772800,    4838400;
   9: 1, 2558, 221958,  4601840,  39198760,  171672480,  429216480,  638184960,   558714240,  266112000,   53222400;
  10: 1, 5118, 673548, 19295192, 219003000, 1265227440, 4206222720, 8539211520, 10772092800, 8248262400, 3512678400, 638668800;
%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) + (x+3*x^2+3*x^3+x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1 + 8*x + 8*x^2.
%t Mathematica
P[1, x_] := 1 + 8*x + 8*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((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, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1 + 8*x + 8*x^2; for(n = 2, N, my(i = n); v[i] = ((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, 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