Real-rooted atlas

A123519 → reverse → D

nonnegative coefficients

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

Recurrence

\(\left(1 + 3\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(6 + 2\,x + 10\,n + 11\,n x + 4\,n^{2} + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x - 6\,x^{2} - 2\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 6 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
0628
12024347
256120604240
314444842012051137
435214402016112021065144
58324224792067202520336722559
61920116482745631680184805040504896736
7435230720873601281281029604435292407209407841

All 15 rows

Technical details
Node id
n-fa59d237e613de64 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-e63306eb47cd27c0

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:     6,      2;
  1:    20,     24,      3;
  2:    56,    120,     60,       4;
  3:   144,    448,    420,     120,       5;
  4:   352,   1440,   2016,    1120,     210,       6;
  5:   832,   4224,   7920,    6720,    2520,     336,      7;
  6:  1920,  11648,  27456,   31680,   18480,    5040,    504,      8;
  7:  4352,  30720,  87360,  128128,  102960,   44352,   9240,    720,     9;
  8:  9728,  78336, 261120,  465920,  480480,  288288,  96096,  15840,   990,   10;
  9: 21504, 194560, 744192, 1566720, 1980160, 1537536, 720720, 192192, 25740, 1320, 11;
%F Formula
The row polynomials satisfy (2*n^2+3*n+1)*P(n,x) = (4*n^2+10*n+6+(6*n^2+11*n+2)*x)*P(n-1,x) - ((2*n+3)*x+(4*n+6)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 6 + 2*x.
%t Mathematica
P[0, x_] := 6 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2 + 10*n + 6 + (6*n^2 + 11*n + 2)*x)*P[n - 1, x] - ((2*n + 3)*x + (4*n + 6)*x^2)*D[P[n - 1, x], x])/(2*n^2 + 3*n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 6 + 2*x; for(n = 1, N, my(i = n + 1); v[i] = ((4*n^2 + 10*n + 6 + (6*n^2 + 11*n + 2)*x)*v[i - 1] - ((2*n + 3)*x + (4*n + 6)*x^2)*deriv(v[i - 1], x))/(2*n^2 + 3*n + 1)); 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. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  4. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023