Real-rooted atlas

A081578 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + x^{2} + 2\,n - 2\,n x - n x^{2} + 2\,n^{2} x\right)\,P_{n}(x) = \left(-22 - 56\,x - 29\,x^{2} - 4\,x^{3} + 34\,n + 74\,n x + 19\,n x^{2} - 8\,n^{2} - 14\,n^{2} x\right)\,P_{n-1}(x) + \left(-8 - 36\,x - 50\,x^{2} - 21\,x^{3} - 2\,x^{4} + 8\,n + 32\,n x + 34\,n x^{2} + 4\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
214822
3609624180
4244624408641340
59103232362414081609334
63282149042378415936432038462610
711480635521322401285125952012288896408488

All 15 rows

Technical details
Node id
n-55fdfd5d28ab774c · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-3e4ae2cc54952178

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:     14,       8;
  3:     60,      96,      24;
  4:    244,     624,     408,      64;
  5:    910,    3232,    3624,    1408,     160;
  6:   3282,   14904,   23784,   15936,    4320,     384;
  7:  11480,   63552,  132240,  128512,   59520,   12288,    896;
  8:  39368,  257120,  660624,  847744,  572800,  199680,  33152,  2048;
  9: 132858, 1000128, 3061296, 4900608, 4407360, 2239488, 620928, 86016, 4608;
%F Formula
The row polynomials satisfy (2*n-2+(2*n^2-2*n)*x-(n-1)*x^2)*P(n,x) = -(8*n^2-34*n+22+(14*n^2-74*n+56)*x-(19*n-29)*x^2+4*x^3)*P(n-1,x) + (8*n-8+(32*n-36)*x+(34*n-50)*x^2+(4*n-21)*x^3-2*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[(-(8*n^2 - 34*n + 22 + (14*n^2 - 74*n + 56)*x - (19*n - 29)*x^2 + 4*x^3)*P[n - 1, x] + (8*n - 8 + (32*n - 36)*x + (34*n - 50)*x^2 + (4*n - 21)*x^3 - 2*x^4)*D[P[n - 1, x], x])/(2*n - 2 + (2*n^2 - 2*n)*x - (n - 1)*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] = (-(8*n^2 - 34*n + 22 + (14*n^2 - 74*n + 56)*x - (19*n - 29)*x^2 + 4*x^3)*v[i - 1] + (8*n - 8 + (32*n - 36)*x + (34*n - 50)*x^2 + (4*n - 21)*x^3 - 2*x^4)*deriv(v[i - 1], x))/(2*n - 2 + (2*n^2 - 2*n)*x - (n - 1)*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.