Real-rooted atlas

A154227 → D

nonnegative coefficients

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

Recurrence

\(\left(8 + 4\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(4\,n^{2} x^{2} + 3\,n^{3} x^{2} - 3\,n^{4} x^{2} - 3\,n^{5} x^{2} - n^{6} x^{2}\right)\,P_{n-4}(x) + \left(-16\,n x - 12\,n x^{2} - 28\,n^{2} x - 20\,n^{2} x^{2} - 16\,n^{3} x - 12\,n^{3} x^{2} - 4\,n^{4} x - 4\,n^{4} x^{2}\right)\,P_{n-3}(x) + \left(-16 - 16\,x - 8\,x^{2} - 12\,n + 8\,n x - 4\,n x^{2} - 4\,n^{2} + 20\,n^{2} x - 4\,n^{2} x^{2} + 16\,n^{3} x + 4\,n^{4} x\right)\,P_{n-2}(x) + \left(24 + 16\,x + 16\,n + 8\,n x + 8\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 8 + 2\,x\)
\(P_{3}(x) = 19 + 38\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
111
28210
31938360
4353161054460
5571184177622853250
6863258168246516430627120
7123753485647114196188357387227080
816915520316863123783252810546560118382146240
9225293889616987941096992637019233961028581800920886840
1029252146254226037107304121973880556609565931940208584262810223480000
113718783060588751405135769426148701131137844245898758161570003952353710112482868080
124631417161330854945705295653392126701558182338474748977389141059123992564770858050931229821182720
1356922055227354348132304018824352935715140758352208164218077576389646971442976840423911811601213036682813372713919760
146903328105319405634888804929456467935093882849954625098914289121251771332728170216422830872220123019504487219968608970144977734023360
Technical details
Node id
n-a0ce9f90b2240ebe · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-651f404dbbd1db01

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:   8,     2;
  3:  19,    38,      3;
  4:  35,   316,    105,       4;
  5:  57,  1184,   1776,     228,       5;
  6:  86,  3258,  16824,    6516,     430,       6;
  7: 123,  7534,  85647,  114196,   18835,     738,      7;
  8: 169, 15520, 316863, 1237832,  528105,   46560,   1183,    8;
  9: 225, 29388, 961698, 7941096, 9926370, 1923396, 102858, 1800, 9;
%F Formula
The row polynomials satisfy (4*n^2+4*n+8)*P(n,x) = -(n^6+3*n^5+3*n^4-3*n^3-4*n^2)*x^2*P(n-4,x) - ((4*n^4+16*n^3+28*n^2+16*n)*x+(4*n^4+12*n^3+20*n^2+12*n)*x^2)*P(n-3,x) - (4*n^2+12*n+16-(4*n^4+16*n^3+20*n^2+8*n-16)*x+(4*n^2+4*n+8)*x^2)*P(n-2,x) + (8*n^2+16*n+24+(8*n^2+8*n+16)*x)*P(n-1,x) for n >= 4, with P(0,x) = 0, P(1,x) = 1, P(2,x) = 8 + 2*x, P(3,x) = 19 + 38*x + 3*x^2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[2, x_] := 8 + 2*x;
P[3, x_] := 19 + 38*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^6 + 3*n^5 + 3*n^4 - 3*n^3 - 4*n^2)*x^2*P[n - 4, x] - ((4*n^4 + 16*n^3 + 28*n^2 + 16*n)*x + (4*n^4 + 12*n^3 + 20*n^2 + 12*n)*x^2)*P[n - 3, x] - (4*n^2 + 12*n + 16 - (4*n^4 + 16*n^3 + 20*n^2 + 8*n - 16)*x + (4*n^2 + 4*n + 8)*x^2)*P[n - 2, x] + (8*n^2 + 16*n + 24 + (8*n^2 + 8*n + 16)*x)*P[n - 1, x])/(4*n^2 + 4*n + 8)]];
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; v[3] = 8 + 2*x; v[4] = 19 + 38*x + 3*x^2; for(n = 4, N, my(i = n + 1); v[i] = (-(n^6 + 3*n^5 + 3*n^4 - 3*n^3 - 4*n^2)*x^2*v[i - 4] - ((4*n^4 + 16*n^3 + 28*n^2 + 16*n)*x + (4*n^4 + 12*n^3 + 20*n^2 + 12*n)*x^2)*v[i - 3] - (4*n^2 + 12*n + 16 - (4*n^4 + 16*n^3 + 20*n^2 + 8*n - 16)*x + (4*n^2 + 4*n + 8)*x^2)*v[i - 2] + (8*n^2 + 16*n + 24 + (8*n^2 + 8*n + 16)*x)*v[i - 1])/(4*n^2 + 4*n + 8)); 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.