Real-rooted atlas

A156366 → D

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(-4 - 2\,n - 12\,n x - 18\,n x^{2} - 6\,n^{2} x - 27\,n^{2} x^{2} - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-2\,x + 18\,x^{3} - n x - 6\,n x^{2} + 27\,n x^{3} - 3\,n^{2} x^{2} + 9\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(4 + 6\,x + 3\,n + 12\,n x + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(x - 3\,x^{2} + n x - 3\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 12 + 18\,x\)
\(P_{1}(x) = 33 + 198\,x + 81\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
0121830
13319881312
278118821063243696
317154362446218468121549752
4360214381956963858841458004374753552
57417727412651395060556521599510803781530912715392
6150626294471469545060556010720431063895392768511852488236854272
730398611203687055242455469615920801101991009808732382560531703441771474831109376

All 15 rows

Technical details
Node id
n-a608f49c6fcfaa74 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-447106697bbaa40b

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:    12,      18;
  1:    33,     198,        81;
  2:    78,    1188,      2106,         324;
  3:   171,    5436,     24462,       18468,         1215;
  4:   360,   21438,    195696,      385884,       145800,         4374;
  5:   741,   77274,   1265139,     5060556,      5215995,      1080378,         15309;
  6:  1506,  262944,   7146954,    50605560,    107204310,     63895392,       7685118,        52488;
  7:  3039,  861120,  36870552,   424554696,   1592080110,   1991009808,     732382560,     53170344,      177147;
  8:  6108, 2747466, 178482528,  3155148936,  19104961320,  42594510636,   33733197792,   8011610856,   360671292,     590490;
  9: 12249, 8608878, 825202485, 21487185576, 197452427490, 710828738964, 1015269518466, 534731210280, 84724272837, 2410970670, 1948617;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = -(2*n+4+(6*n^2+12*n)*x+(9*n^3+27*n^2+18*n)*x^2)*P(n-2,x) - ((n+2)*x+(3*n^2+6*n)*x^2-(9*n^2+27*n+18)*x^3)*d/dx P(n-2,x) + (3*n+4+(6*n^2+12*n+6)*x)*P(n-1,x) + ((n+1)*x-(3*n+3)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 12 + 18*x, P(1,x) = 33 + 198*x + 81*x^2.
%t Mathematica
P[0, x_] := 12 + 18*x;
P[1, x_] := 33 + 198*x + 81*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 4 + (6*n^2 + 12*n)*x + (9*n^3 + 27*n^2 + 18*n)*x^2)*P[n - 2, x] - ((n + 2)*x + (3*n^2 + 6*n)*x^2 - (9*n^2 + 27*n + 18)*x^3)*D[P[n - 2, x], x] + (3*n + 4 + (6*n^2 + 12*n + 6)*x)*P[n - 1, x] + ((n + 1)*x - (3*n + 3)*x^2)*D[P[n - 1, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 12 + 18*x; v[2] = 33 + 198*x + 81*x^2; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n + 4 + (6*n^2 + 12*n)*x + (9*n^3 + 27*n^2 + 18*n)*x^2)*v[i - 2] - ((n + 2)*x + (3*n^2 + 6*n)*x^2 - (9*n^2 + 27*n + 18)*x^3)*deriv(v[i - 2], x) + (3*n + 4 + (6*n^2 + 12*n + 6)*x)*v[i - 1] + ((n + 1)*x - (3*n + 3)*x^2)*deriv(v[i - 1], x))/(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

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.