Real-rooted atlas

A154233 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
219919
31226040123
41432945022511091
5174104734684155166210407
61117299919379437274084513615120683
711747358870323308105475644152401186401506819
81247160843229842093712568971075149804815780120394521657443
913383215310255001114039047065072970696861055352965835082512966850333186111
101449598612869885501498023327657161039592828173898587816134742577858185251571637055720880907
111582105168724592819444211520267969109330039782226998248323163269865525481848890110714161802012984760104457403803
121739176082168043146630446011063373262572735538892254075478716507712047749608378567125433925495170170318062770283863437952086845292579
131922283083362972282027201605484653576104960472676672455133997752681326850031511457989413770119884157350657654310030780273729903947542112292915044074116054423
1411133439587738189835648698825194608184055295400442312120871406280139800270391758911842981194455852681673330712712494726905205651443328227104954742494198018567749917114551004523680461291
Technical details
Node id
n-3c6f163a8ebc0747 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ec0cbe08708f3ce3

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,   9,     9;
  3: 1,  22,    60,      40;
  4: 1,  43,   294,     502,      251;
  5: 1,  74,  1047,    3468,     4155,     1662;
  6: 1, 117,  2999,   19379,    43727,    40845,    13615;
  7: 1, 174,  7358,   87032,   330810,   547564,   415240,    118640;
  8: 1, 247, 16084,  322984,  2093712,  5689710,  7514980,   4815780,  1203945;
  9: 1, 338, 32153, 1025500, 11140390, 47065072, 97069686, 105535296, 58350825, 12966850;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = ((n^2+n-1)*x+(2*n^2+2*n-2)*x^2+(n^2+n-1)*x^3)*P(n-2,x) + (1+3*x+2*x^2)*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[(((n^2 + n - 1)*x + (2*n^2 + 2*n - 2)*x^2 + (n^2 + n - 1)*x^3)*P[n - 2, x] + (1 + 3*x + 2*x^2)*P[n - 1, x])/(1 + x)]];
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] = (((n^2 + n - 1)*x + (2*n^2 + 2*n - 2)*x^2 + (n^2 + n - 1)*x^3)*v[i - 2] + (1 + 3*x + 2*x^2)*v[i - 1])/(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