Real-rooted atlas

A126216 → basis A013610 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(6 + 12\,x + 6\,x^{2} + 3\,n + 6\,n x + 3\,n x^{2}\right)\,P_{n}(x) = \left(2 - 16\,x - 78\,x^{2} - 100\,x^{3} - 40\,x^{4} + 25\,n + 124\,n x + 213\,n x^{2} + 154\,n x^{3} + 40\,n x^{4}\right)\,P_{n-1}(x) + \left(-4 - 38\,x - 130\,x^{2} - 202\,x^{3} - 146\,x^{4} - 40\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1369
211434195
3452614953061107
4197151442965324242613757
5903864032640607805571020076178749
64279489932311355745957930955756731714812399251
72079327717915681334877565900196598504915912823150066633009615

All 15 rows

Technical details
Node id
n-503a4b66c4a2e621 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-08e302df1914bf50

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:      3,       6;
  2:     11,      43,       41;
  3:     45,     261,      495,       306;
  4:    197,    1514,     4296,      5324,      2426;
  5:    903,    8640,    32640,     60780,     55710,      20076;
  6:   4279,   48993,   231135,    574595,    793095,     575673,     171481;
  7:  20793,  277179,  1568133,   4877565,   9001965,    9850491,    5912823,    1500666;
  8: 103049, 1567444, 10340624,  38625244,  89299574,  130777780,  118403936,   60556360,  13386206;
  9: 518859, 8867610, 66832560, 291419100, 809859330, 1486838808, 1802518704, 1390770216, 619414074, 121267476;
%F Formula
The row polynomials satisfy (3*n+6+(6*n+12)*x+(3*n+6)*x^2)*P(n,x) = (25*n+2+(124*n-16)*x+(213*n-78)*x^2+(154*n-100)*x^3+(40*n-40)*x^4)*P(n-1,x) - (4+38*x+130*x^2+202*x^3+146*x^4+40*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[((25*n + 2 + (124*n - 16)*x + (213*n - 78)*x^2 + (154*n - 100)*x^3 + (40*n - 40)*x^4)*P[n - 1, x] - (4 + 38*x + 130*x^2 + 202*x^3 + 146*x^4 + 40*x^5)*D[P[n - 1, x], x])/(3*n + 6 + (6*n + 12)*x + (3*n + 6)*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] = ((25*n + 2 + (124*n - 16)*x + (213*n - 78)*x^2 + (154*n - 100)*x^3 + (40*n - 40)*x^4)*v[i - 1] - (4 + 38*x + 130*x^2 + 202*x^3 + 146*x^4 + 40*x^5)*deriv(v[i - 1], x))/(3*n + 6 + (6*n + 12)*x + (3*n + 6)*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

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