Real-rooted atlas

A207823 → reverse → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(2\,n\right)\,P_{n}(x) = \left(-16 - 28\,x - 12\,x^{2} + 26\,n + 36\,n x + 12\,n x^{2}\right)\,P_{n-1}(x) + \left(-21 - 53\,x - 44\,x^{2} - 12\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1549
224381577
311527221456657
45511734204310682095605
52640103721627812756499178047817
6126495958411681612200471591223782911407933
760605332864782754102160079919037473697516108643480129

All 15 rows

Technical details
Node id
n-74e8b5eac5b04b50 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-b0bdfa529c7b31c5

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:       5,       4;
  2:      24,      38,       15;
  3:     115,     272,      214,       56;
  4:     551,    1734,     2043,     1068,      209;
  5:    2640,   10372,    16278,    12756,     4991,      780;
  6:   12649,   59584,   116816,   122004,    71591,    22378,     2911;
  7:   60605,  332864,   782754,  1021600,   799190,   374736,    97516,   10864;
  8:  290376, 1821858,  4996593,  7823796,  7649935,  4782876,  1867266,  416184,   40545;
  9: 1391275, 9816720, 30761310, 56185380, 65919715, 51519404, 26821656, 8969352, 1748217, 151316;
%F Formula
The row polynomials satisfy 2*n*P(n,x) = (26*n-16+(36*n-28)*x+(12*n-12)*x^2)*P(n-1,x) - (21+53*x+44*x^2+12*x^3)*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[((26*n - 16 + (36*n - 28)*x + (12*n - 12)*x^2)*P[n - 1, x] - (21 + 53*x + 44*x^2 + 12*x^3)*D[P[n - 1, x], x])/(2*n)]];
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] = ((26*n - 16 + (36*n - 28)*x + (12*n - 12)*x^2)*v[i - 1] - (21 + 53*x + 44*x^2 + 12*x^3)*deriv(v[i - 1], x))/(2*n)); 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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023