Real-rooted atlas

A113413 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(4 + 5\,x + 2\,x^{2} - 6\,n - 4\,n x + 2\,n^{2} - 2\,n^{2} x - 2\,n^{2} x^{2} + n^{3} x\right)\,P_{n}(x) = \left(8 + 16\,x + 4\,x^{2} - 28\,n - 75\,n x - 54\,n x^{2} - 12\,n x^{3} + 16\,n^{2} + 35\,n^{2} x + 14\,n^{2} x^{2} - 2\,n^{3} - 3\,n^{3} x\right)\,P_{n-1}(x) + \left(2 + 8\,x + 9\,x^{2} + 2\,x^{3} - 6\,n - 28\,n x - 43\,n x^{2} - 24\,n x^{3} - 4\,n x^{4} + 2\,n^{2} + 8\,n^{2} x + 9\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
212820
3467224142
415238428864888
5466163219689601605186
613646120102728000288038429020
73870211924576849600283208064896157710

All 15 rows

Technical details
Node id
n-fa10445850d8425c · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-19ba0a1360e0730c

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:     2;
  2:    12,      8;
  3:    46,     72,     24;
  4:   152,    384,    288,      64;
  5:   466,   1632,   1968,     960,     160;
  6:  1364,   6120,  10272,    8000,    2880,    384;
  7:  3870,  21192,  45768,   49600,   28320,   8064,    896;
  8: 10736,  69440, 183744,  256640,  203520,  91392,  21504,  2048;
  9: 29282, 218496, 685152, 1174912, 1200960, 747264, 275968, 55296, 4608;
%F Formula
The row polynomials satisfy (2*n^2-6*n+4+(n^3-2*n^2-4*n+5)*x-(2*n^2-2)*x^2)*P(n,x) = -(2*n^3-16*n^2+28*n-8+(3*n^3-35*n^2+75*n-16)*x-(14*n^2-54*n+4)*x^2+12*n*x^3)*P(n-1,x) + (2*n^2-6*n+2+(8*n^2-28*n+8)*x+(9*n^2-43*n+9)*x^2+(2*n^2-24*n+2)*x^3-4*n*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^3 - 16*n^2 + 28*n - 8 + (3*n^3 - 35*n^2 + 75*n - 16)*x - (14*n^2 - 54*n + 4)*x^2 + 12*n*x^3)*P[n - 1, x] + (2*n^2 - 6*n + 2 + (8*n^2 - 28*n + 8)*x + (9*n^2 - 43*n + 9)*x^2 + (2*n^2 - 24*n + 2)*x^3 - 4*n*x^4)*D[P[n - 1, x], x])/(2*n^2 - 6*n + 4 + (n^3 - 2*n^2 - 4*n + 5)*x - (2*n^2 - 2)*x^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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(2*n^3 - 16*n^2 + 28*n - 8 + (3*n^3 - 35*n^2 + 75*n - 16)*x - (14*n^2 - 54*n + 4)*x^2 + 12*n*x^3)*v[i - 1] + (2*n^2 - 6*n + 2 + (8*n^2 - 28*n + 8)*x + (9*n^2 - 43*n + 9)*x^2 + (2*n^2 - 24*n + 2)*x^3 - 4*n*x^4)*deriv(v[i - 1], x))/(2*n^2 - 6*n + 4 + (n^3 - 2*n^2 - 4*n + 5)*x - (2*n^2 - 2)*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.