Real-rooted atlas

A122848 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-6 - 36\,x - 54\,x^{2} - 45\,n - 54\,n x + 27\,n^{2}\right)\,P_{n}(x) = \left(-84 - 360\,x - 378\,x^{2} - 162\,x^{3} - 138\,n - 207\,n x - 108\,n x^{2} + 153\,n^{2} + 135\,n^{2} x - 27\,n^{3}\right)\,P_{n-1}(x) + \left(-16 - 108\,x - 216\,x^{2} - 108\,x^{3} - 30\,n - 126\,n x - 108\,n x^{2} + 18\,n^{2} + 54\,n^{2} x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1437
21127947
33113513527328
4945701053594812392
5296234063006615243024318224
699495583388553595364509477729144688
73466395221740693694953855601837083572121871193728

All 15 rows

Technical details
Node id
n-9496a11d823f8ed4 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-384f7c1322da5148

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:     4,      3;
  2:    11,     27,       9;
  3:    31,    135,     135,       27;
  4:    94,    570,    1053,      594,       81;
  5:   296,   2340,    6300,     6615,     2430,      243;
  6:   994,   9558,   33885,    53595,    36450,     9477,      729;
  7:  3466,  39522,  174069,   369495,   385560,   183708,    35721,    2187;
  8: 12716, 166680,  875700,  2341332,  3342465,  2463048,   867510,  131220,   6561;
  9: 48304, 717660, 4384368, 14147784, 25780356, 26377407, 14390460, 3897234, 472392, 19683;
%F Formula
The row polynomials satisfy (27*n^2-45*n-6-(54*n+36)*x-54*x^2)*P(n,x) = -(27*n^3-153*n^2+138*n+84-(135*n^2-207*n-360)*x+(108*n+378)*x^2+162*x^3)*P(n-1,x) + (18*n^2-30*n-16+(54*n^2-126*n-108)*x-(108*n+216)*x^2-108*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[(-(27*n^3 - 153*n^2 + 138*n + 84 - (135*n^2 - 207*n - 360)*x + (108*n + 378)*x^2 + 162*x^3)*P[n - 1, x] + (18*n^2 - 30*n - 16 + (54*n^2 - 126*n - 108)*x - (108*n + 216)*x^2 - 108*x^3)*D[P[n - 1, x], x])/(27*n^2 - 45*n - 6 - (54*n + 36)*x - 54*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] = (-(27*n^3 - 153*n^2 + 138*n + 84 - (135*n^2 - 207*n - 360)*x + (108*n + 378)*x^2 + 162*x^3)*v[i - 1] + (18*n^2 - 30*n - 16 + (54*n^2 - 126*n - 108)*x - (108*n + 216)*x^2 - 108*x^3)*deriv(v[i - 1], x))/(27*n^2 - 45*n - 6 - (54*n + 36)*x - 54*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. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.