Real-rooted atlas

A110441 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(114\,n + 18\,n x - 54\,n x^{2} + 54\,n^{3}\right)\,P_{n}(x) = \left(264 + 72\,x - 162\,x^{2} - 81\,x^{3} + 480\,n + 234\,n x - 162\,n x^{2} - 81\,n x^{3} + 324\,n^{2} + 243\,n^{2} x + 108\,n^{3} + 81\,n^{3} x\right)\,P_{n-1}(x) + \left(32 - 24\,x - 396\,x^{2} - 351\,x^{3} - 81\,x^{4} + 72\,n + 216\,n x + 81\,n x^{2} + 72\,n^{2} + 216\,n^{2} x + 81\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
17310
23842989
318034818927744
478822442106756815975
53284125161720810584283524346670
613240636001168561047604779010206729357181
7521283030727022168391605556602012043572121872691348

All 15 rows

Technical details
Node id
n-5e244dd8076141b2 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-43f5521f5a12e7be

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:      7,       3;
  2:     38,      42,        9;
  3:    180,     348,      189,       27;
  4:    788,    2244,     2106,      756,       81;
  5:   3284,   12516,    17208,    10584,     2835,      243;
  6:  13240,   63600,   116856,   104760,    47790,    10206,      729;
  7:  52128,  303072,   702216,   839160,   555660,   201204,    35721,    2187;
  8: 201616, 1378272,  3870576,  5827248,  5130540,  2687580,   806274,  122472,   6561;
  9: 769072, 6049776, 20011104, 36513504, 40445244, 28107324, 12165552, 3114288, 413343, 19683;
%F Formula
The row polynomials satisfy (54*n^3+114*n+18*n*x-54*n*x^2)*P(n,x) = (108*n^3+324*n^2+480*n+264+(81*n^3+243*n^2+234*n+72)*x-(162*n+162)*x^2-(81*n+81)*x^3)*P(n-1,x) + (72*n^2+72*n+32+(216*n^2+216*n-24)*x+(81*n^2+81*n-396)*x^2-351*x^3-81*x^4)*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[((108*n^3 + 324*n^2 + 480*n + 264 + (81*n^3 + 243*n^2 + 234*n + 72)*x - (162*n + 162)*x^2 - (81*n + 81)*x^3)*P[n - 1, x] + (72*n^2 + 72*n + 32 + (216*n^2 + 216*n - 24)*x + (81*n^2 + 81*n - 396)*x^2 - 351*x^3 - 81*x^4)*D[P[n - 1, x], x])/(54*n^3 + 114*n + 18*n*x - 54*n*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] = ((108*n^3 + 324*n^2 + 480*n + 264 + (81*n^3 + 243*n^2 + 234*n + 72)*x - (162*n + 162)*x^2 - (81*n + 81)*x^3)*v[i - 1] + (72*n^2 + 72*n + 32 + (216*n^2 + 216*n - 24)*x + (81*n^2 + 81*n - 396)*x^2 - 351*x^3 - 81*x^4)*deriv(v[i - 1], x))/(54*n^3 + 114*n + 18*n*x - 54*n*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. [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