Real-rooted atlas

A257617 → D

nonnegative coefficients

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

Recurrence

\(\left(-12 + 7\,n\right)\,P_{n}(x) = \left(90 - 380\,x + 1140\,x^{2} - 126\,n + 672\,n x - 2681\,n x^{2} - 196\,n^{2} x + 1764\,n^{2} x^{2} - 343\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(70\,x - 490\,x^{2} + 420\,x^{3} - 98\,n x + 931\,n x^{2} - 833\,n x^{3} - 343\,n^{2} x^{2} + 343\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-118 + 288\,x + 77\,n - 336\,n x + 98\,n^{2} x\right)\,P_{n-1}(x) + \left(-84\,x + 84\,x^{2} + 49\,n x - 49\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
236844
3388776241188
4367624832110286439600
5335645664088496121342561601584000
6303260110763683908215222152736151630038474131200
7273215619966312813973260561863101408499157820163929368963978374400

All 15 rows

Technical details
Node id
n-3618a761aebd57c3 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-0c5574eb81f1a737

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:        36,           8;
  3:       388,         776,            24;
  4:      3676,       24832,         11028,            64;
  5:     33564,      566408,        849612,        134256,           160;
  6:    303260,    11076368,      39082152,      22152736,       1516300,           384;
  7:   2732156,   199663128,    1397326056,    1863101408,     499157820,      16392936,          896;
  8:  24595932,  3435039776,   43219802892,  111786084480,   72033004820,   10305119328,    172171524,       2048;
  9: 221378236, 57469421480, 1220768091732, 5485759477072, 6857199346340, 2441536183464, 201142975180, 1771025888, 4608;
%F Formula
The row polynomials satisfy (7*n-12)*P(n,x) = -(126*n-90+(196*n^2-672*n+380)*x+(343*n^3-1764*n^2+2681*n-1140)*x^2)*P(n-2,x) - ((98*n-70)*x+(343*n^2-931*n+490)*x^2-(343*n^2-833*n+420)*x^3)*d/dx P(n-2,x) + (77*n-118+(98*n^2-336*n+288)*x)*P(n-1,x) + ((49*n-84)*x-(49*n-84)*x^2)*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[(-(126*n - 90 + (196*n^2 - 672*n + 380)*x + (343*n^3 - 1764*n^2 + 2681*n - 1140)*x^2)*P[n - 2, x] - ((98*n - 70)*x + (343*n^2 - 931*n + 490)*x^2 - (343*n^2 - 833*n + 420)*x^3)*D[P[n - 2, x], x] + (77*n - 118 + (98*n^2 - 336*n + 288)*x)*P[n - 1, x] + ((49*n - 84)*x - (49*n - 84)*x^2)*D[P[n - 1, x], x])/(7*n - 12)]];
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] = (-(126*n - 90 + (196*n^2 - 672*n + 380)*x + (343*n^3 - 1764*n^2 + 2681*n - 1140)*x^2)*v[i - 2] - ((98*n - 70)*x + (343*n^2 - 931*n + 490)*x^2 - (343*n^2 - 833*n + 420)*x^3)*deriv(v[i - 2], x) + (77*n - 118 + (98*n^2 - 336*n + 288)*x)*v[i - 1] + ((49*n - 84)*x - (49*n - 84)*x^2)*deriv(v[i - 1], x))/(7*n - 12)); 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.