Real-rooted atlas

A122848 → basis A007318 → D

nonnegative coefficients

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

Recurrence

\(\left(2 + 2\,x - 3\,n - 2\,n x + n^{2}\right)\,P_{n}(x) = \left(-10\,n - 9\,n x - 2\,n x^{2} + 7\,n^{2} + 3\,n^{2} x - n^{3}\right)\,P_{n-1}(x) + \left(-6\,n - 10\,n x - 4\,n x^{2} + 2\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
2325
3912324
42854304116
590230195605580
6306960109554010563012
710784032577539201260168716240
83984171922956825620116202604252890848
91522874664149940158004929252998849143609526032
1060580331560760140943320683235290430695108640495103147920
1124855615039203876840553608047666852525292807345147840143556601119427584
121055088698095219965000322304403217302020458746824531420394002930402277085812123464640
134606264331035121040290681873128402127024901580101387715607924308856475546554769034749109213806568256
14207128881605007045493579001091073984138883745011811159366789813032615012406573867310360350973973513241365145409207104
Technical details
Node id
n-e13b049d25118e46 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-72f5e45cef73a5a2

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:     1;
  2:     3,     2;
  3:     9,    12,      3;
  4:    28,    54,     30,      4;
  5:    90,   230,    195,     60,     5;
  6:   306,   960,   1095,    540,   105,     6;
  7:  1078,  4032,   5775,   3920,  1260,   168,    7;
  8:  3984, 17192,  29568,  25620, 11620,  2604,  252,   8;
  9: 15228, 74664, 149940, 158004, 92925, 29988, 4914, 360, 9;
%F Formula
The row polynomials satisfy (n^2-3*n+2-(2*n-2)*x)*P(n,x) = -(n^3-7*n^2+10*n-(3*n^2-9*n)*x+2*n*x^2)*P(n-1,x) + (2*n^2-6*n+(2*n^2-10*n)*x-4*n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 7*n^2 + 10*n - (3*n^2 - 9*n)*x + 2*n*x^2)*P[n - 1, x] + (2*n^2 - 6*n + (2*n^2 - 10*n)*x - 4*n*x^2)*D[P[n - 1, x], x])/(n^2 - 3*n + 2 - (2*n - 2)*x)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 7*n^2 + 10*n - (3*n^2 - 9*n)*x + 2*n*x^2)*v[i - 1] + (2*n^2 - 6*n + (2*n^2 - 10*n)*x - 4*n*x^2)*deriv(v[i - 1], x))/(n^2 - 3*n + 2 - (2*n - 2)*x)); 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.