Real-rooted atlas

A257606 → D

nonnegative coefficients

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

Recurrence

\(\left(2 + n\right)\,P_{n}(x) = \left(-60 - 24\,x - 6\,x^{2} - 20\,n - 32\,n x - 11\,n x^{2} - 8\,n^{2} x - 6\,n^{2} x^{2} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-12\,x + 6\,x^{2} + 6\,x^{3} - 4\,n x - n x^{2} + 5\,n x^{3} - n^{2} x^{2} + n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(22 + 8\,x + 9\,n + 8\,n x + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2} + n x - n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
144
2403272
32965921921080
4192871045784102415840
51168869616104424467525120237600
66765660470414619361209408338280245763706560
73792404846032174900002332000012115080227544011468860540480

All 15 rows

Technical details
Node id
n-2ba9a596f53694d1 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-2603e33c60dc5afd

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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:        0;
  1:        4;
  2:       40,        32;
  3:      296,       592,        192;
  4:     1928,      7104,       5784,       1024;
  5:    11688,     69616,     104424,      46752,       5120;
  6:    67656,    604704,    1461936,    1209408,     338280,      24576;
  7:   379240,   4846032,   17490000,   23320000,   12115080,    2275440,    114688;
  8:  2076424,  36660992,  187851432,  373120000,  313085720,  109982976,  14534968,   524288;
  9: 11168552, 265647280, 1864874904, 5239177184, 6548971480, 3729749808, 929765480, 89348416, 2359296;
%F Formula
The row polynomials satisfy (n+2)*P(n,x) = -(20*n+60+(8*n^2+32*n+24)*x+(n^3+6*n^2+11*n+6)*x^2)*P(n-2,x) - ((4*n+12)*x+(n^2+n-6)*x^2-(n^2+5*n+6)*x^3)*d/dx P(n-2,x) + (9*n+22+(2*n^2+8*n+8)*x)*P(n-1,x) + ((n+2)*x-(n+2)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 4.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(20*n + 60 + (8*n^2 + 32*n + 24)*x + (n^3 + 6*n^2 + 11*n + 6)*x^2)*P[n - 2, x] - ((4*n + 12)*x + (n^2 + n - 6)*x^2 - (n^2 + 5*n + 6)*x^3)*D[P[n - 2, x], x] + (9*n + 22 + (2*n^2 + 8*n + 8)*x)*P[n - 1, x] + ((n + 2)*x - (n + 2)*x^2)*D[P[n - 1, x], x])/(n + 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 4; for(n = 2, N, my(i = n + 1); v[i] = (-(20*n + 60 + (8*n^2 + 32*n + 24)*x + (n^3 + 6*n^2 + 11*n + 6)*x^2)*v[i - 2] - ((4*n + 12)*x + (n^2 + n - 6)*x^2 - (n^2 + 5*n + 6)*x^3)*deriv(v[i - 2], x) + (9*n + 22 + (2*n^2 + 8*n + 8)*x)*v[i - 1] + ((n + 2)*x - (n + 2)*x^2)*deriv(v[i - 1], x))/(n + 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.