Real-rooted atlas

A001263 → basis A013609 → x ↦ x/(x+1)

nonnegative coefficients

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

Recurrence

\(\left(2 + 4\,x + 2\,x^{2} + n + 2\,n x + n x^{2}\right)\,P_{n}(x) = \left(2 + 4\,x - 4\,x^{2} - 12\,x^{3} - 6\,x^{4} + 4\,n + 20\,n x + 34\,n x^{2} + 24\,n x^{3} + 6\,n x^{4}\right)\,P_{n-1}(x) + \left(-2\,x - 12\,x^{2} - 24\,x^{3} - 20\,x^{4} - 6\,x^{5}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1246
25201944
31484162100360
44233698012325623152
51321320516098409140330428896
642951482524564680912156706820071273856
71430200201181183803807214908054204896501249962661504

All 15 rows

Technical details
Node id
n-ee880590a3a03de2 · revision 0 · JSON
Definition source
derived:mobius_x_over_x_plus_one
Fingerprint
f6-759efd4d0f2bc3f6

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:     2,      4;
  2:     5,     20,      19;
  3:    14,     84,     162,      100;
  4:    42,    336,     980,     1232,      562;
  5:   132,   1320,    5160,     9840,     9140,     3304;
  6:   429,   5148,   25245,    64680,    91215,    67068,    20071;
  7:  1430,  20020,  118118,   380380,   721490,   805420,   489650,   124996;
  8:  4862,  77792,  536536,  2082080,  4968964,  7463456,  6886264,  3566816,   793774;
  9: 16796, 302328, 2386800, 10841376, 31207176, 59007312, 73255728, 57556512, 25961148, 5120632;
%F Formula
The row polynomials satisfy (n+2+(2*n+4)*x+(n+2)*x^2)*P(n,x) = (4*n+2+(20*n+4)*x+(34*n-4)*x^2+(24*n-12)*x^3+(6*n-6)*x^4)*P(n-1,x) - (2*x+12*x^2+24*x^3+20*x^4+6*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 + (20*n + 4)*x + (34*n - 4)*x^2 + (24*n - 12)*x^3 + (6*n - 6)*x^4)*P[n - 1, x] - (2*x + 12*x^2 + 24*x^3 + 20*x^4 + 6*x^5)*D[P[n - 1, x], x])/(n + 2 + (2*n + 4)*x + (n + 2)*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 + (20*n + 4)*x + (34*n - 4)*x^2 + (24*n - 12)*x^3 + (6*n - 6)*x^4)*v[i - 1] - (2*x + 12*x^2 + 24*x^3 + 20*x^4 + 6*x^5)*deriv(v[i - 1], x))/(n + 2 + (2*n + 4)*x + (n + 2)*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