Real-rooted atlas

A269951 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2011
30235
405151131
50167110349239
60653768037522612257
703262299640688296026163125517
801957161715513499381999745325511743337615
9013700129275518631114761415136811190734517511959015127047
1001096011159880533439813934096226221792338532815037350550103287680987960673
11098641011551011597756121795061303448765264397715693723108282018479146360493888776911683108629
12098641011264863037262700332460009826544529242582329315338612806271615740745028795832797956220079864101135544914239
13010850511215108306399520921895358555446918968699285815649367241119441448026117199295953310620518025843602691901107131750231193556233821298510815
1401302061345195515103201340434394415549855893681545901129618305261685305643825360777654609935730808352668952786619138733621606994448105691545787370321562473614120611083565489
15016926797486272517974299201806774755491035512133272791637160555261464474453238999796920692581216693542881951107874223972247455849784483036050842911370118611032988232380223911586220048367319558299094880061
Technical details
Node id
n-772cd5e88801d525 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-828e173002afaa4c

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: 0,      1;
   3: 0,      2,       3;
   4: 0,      5,      15,      11;
   5: 0,     16,      71,     103,       49;
   6: 0,     65,     376,     803,      752,      261;
   7: 0,    326,    2299,    6406,     8829,     6026,     1631;
   8: 0,   1957,   16171,   55134,    99381,    99974,    53255,    11743;
   9: 0,  13700,  129275,  518631,  1147614,  1513681,  1190734,   517511,   95901;
  10: 0, 109601, 1159880, 5334398, 13934096, 22622179, 23385328, 15037350, 5501032, 876809;
%F Formula
The row polynomials satisfy (1+x)*P(n,x) = -(n-3+(3*n-9)*x+(3*n-9)*x^2+(n-3)*x^3)*P(n-2,x) + (n-1+(2*n-1)*x+n*x^2)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n - 3 + (3*n - 9)*x + (3*n - 9)*x^2 + (n - 3)*x^3)*P[n - 2, x] + (n - 1 + (2*n - 1)*x + n*x^2)*P[n - 1, x])/(1 + x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = x; for(n = 3, N, my(i = n); v[i] = (-(n - 3 + (3*n - 9)*x + (3*n - 9)*x^2 + (n - 3)*x^3)*v[i - 2] + (n - 1 + (2*n - 1)*x + n*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, 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