Real-rooted atlas

A132812 → basis A007318 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2\,n + 4\,n x + 2\,n x^{2} + n^{2} + 2\,n^{2} x + n^{2} x^{2}\right)\,P_{n}(x) = \left(2 + 4\,x - 2\,x^{2} - 8\,x^{3} - 4\,x^{4} + 6\,n + 21\,n x + 24\,n x^{2} + 9\,n x^{3} + 4\,n^{2} + 17\,n^{2} x + 26\,n^{2} x^{2} + 17\,n^{2} x^{3} + 4\,n^{2} x^{4}\right)\,P_{n-1}(x) + \left(-2\,x - 10\,x^{2} - 18\,x^{3} - 14\,x^{4} - 4\,x^{5} - 2\,n x - 10\,n x^{2} - 18\,n x^{3} - 14\,n x^{4} - 4\,n x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
14610
215453393
356252372180860
42101260280027309857985
57925940176402592018840541874550
630032702710048519750521640512530729953699685
7114401201205365361321320193732016909208134001663446597400

All 15 rows

Technical details
Node id
n-2bcf501d4853538e · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-ca6d936c14642db2

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:      4,       6;
  2:     15,      45,       33;
  3:     56,     252,      372,      180;
  4:    210,    1260,     2800,     2730,       985;
  5:    792,    5940,    17640,    25920,     18840,      5418;
  6:   3003,   27027,   100485,   197505,    216405,    125307,     29953;
  7:  11440,  120120,   536536,  1321320,   1937320,   1690920,    813400,   166344;
  8:  43758,  525096,  2738736,  8108100,  14900886,  17405388,  12618144,  5190372,   927441;
  9: 167960, 2267460, 13525200, 46781280, 103390560, 151400340, 146884920, 91035360, 32704380, 5188590;
%F Formula
The row polynomials satisfy (n^2+2*n+(2*n^2+4*n)*x+(n^2+2*n)*x^2)*P(n,x) = (4*n^2+6*n+2+(17*n^2+21*n+4)*x+(26*n^2+24*n-2)*x^2+(17*n^2+9*n-8)*x^3+(4*n^2-4)*x^4)*P(n-1,x) - ((2*n+2)*x+(10*n+10)*x^2+(18*n+18)*x^3+(14*n+14)*x^4+(4*n+4)*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 6*n + 2 + (17*n^2 + 21*n + 4)*x + (26*n^2 + 24*n - 2)*x^2 + (17*n^2 + 9*n - 8)*x^3 + (4*n^2 - 4)*x^4)*P[n - 1, x] - ((2*n + 2)*x + (10*n + 10)*x^2 + (18*n + 18)*x^3 + (14*n + 14)*x^4 + (4*n + 4)*x^5)*D[P[n - 1, x], x])/(n^2 + 2*n + (2*n^2 + 4*n)*x + (n^2 + 2*n)*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; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 6*n + 2 + (17*n^2 + 21*n + 4)*x + (26*n^2 + 24*n - 2)*x^2 + (17*n^2 + 9*n - 8)*x^3 + (4*n^2 - 4)*x^4)*v[i - 1] - ((2*n + 2)*x + (10*n + 10)*x^2 + (18*n + 18)*x^3 + (14*n + 14)*x^4 + (4*n + 4)*x^5)*deriv(v[i - 1], x))/(n^2 + 2*n + (2*n^2 + 4*n)*x + (n^2 + 2*n)*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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023