Real-rooted atlas

A125662 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

\(\left(2\,n\right)\,P_{n}(x) = \left(-4\,x - 4\,x^{2} + 6\,n + 12\,n x + 4\,n x^{2}\right)\,P_{n-1}(x) + \left(-3\,x - 8\,x^{2} - 4\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
29241548
32710813856329
4814328376962092255
5243162042125328327378015456
67295832190353240030294147362911105937
7218720412801901717202162701600566440410864726103

All 15 rows

Technical details
Node id
n-6473bfeb7fc71c06 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-5f3ddac121e398b5

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,      4;
  2:     9,     24,      15;
  3:    27,    108,     138,      56;
  4:    81,    432,     837,     696,     209;
  5:   243,   1620,    4212,    5328,    3273,     780;
  6:   729,   5832,   19035,   32400,   30294,   14736,    2911;
  7:  2187,  20412,   80190,  171720,  216270,  160056,   64404,   10864;
  8:  6561,  69984,  321489,  830088, 1316655, 1312848,  803142,  275472,   40545;
  9: 19683, 236196, 1242216, 3755808, 7190127, 9033444, 7444548, 3878928, 1159119, 151316;
%F Formula
The row polynomials satisfy 2*n*P(n,x) = (6*n+(12*n-4)*x+(4*n-4)*x^2)*P(n-1,x) - (3*x+8*x^2+4*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[((6*n + (12*n - 4)*x + (4*n - 4)*x^2)*P[n - 1, x] - (3*x + 8*x^2 + 4*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] = ((6*n + (12*n - 4)*x + (4*n - 4)*x^2)*v[i - 1] - (3*x + 8*x^2 + 4*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. [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. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023