Real-rooted atlas

A375853 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-7 + 9\,x - 2\,x^{2} + 12\,n - 13\,n x + 2\,n x^{2} - 7\,n^{2} + 4\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(8 - 22\,x + 18\,x^{2} - 4\,x^{3} + 2\,n + 15\,n x - 27\,n x^{2} + 6\,n x^{3} + 3\,n^{2} - 11\,n^{2} x + 12\,n^{2} x^{2} + 2\,n^{3} + 6\,n^{3} x\right)\,P_{n-1}(x) + \left(6\,x - 16\,x^{2} + 14\,x^{3} - 4\,x^{4} - 4\,n x + 12\,n x^{2} - 8\,n x^{3} + 4\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
122
216824
3769620192
4256648336401280
568629923036896707680
61568106081830410400201611243008
73192311048268079040291604032168229376
85952791523019204420002720007099273922401179648

All 15 rows

Technical details
Node id
n-63ccdddb32d321a1 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-ea07d872e5cb1eb5

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     2;
   2:    16,      8;
   3:    76,     96,      20;
   4:   256,    648,     336,      40;
   5:   686,   2992,    3036,     896,      70;
   6:  1568,  10608,   18304,   10400,    2016,     112;
   7:  3192,  31104,   82680,   79040,   29160,    4032,     168;
   8:  5952,  79152,  301920,  442000,  272000,   70992,    7392,    240;
   9: 10362, 180576,  937992, 1963840, 1841100,  795872,  155496,  12672,   330;
  10: 17072, 377784, 2568192, 7307552, 9767520, 6403152, 2059904, 313632, 20592, 440;
%F Formula
The row polynomials satisfy (2*n^3-7*n^2+12*n-7+(4*n^2-13*n+9)*x+(2*n-2)*x^2)*P(n,x) = (2*n^3+3*n^2+2*n+8+(6*n^3-11*n^2+15*n-22)*x+(12*n^2-27*n+18)*x^2+(6*n-4)*x^3)*P(n-1,x) + ((4*n^2-4*n+6)*x-(4*n^2-12*n+16)*x^2-(8*n-14)*x^3-4*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 2.
%t Mathematica
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 + 3*n^2 + 2*n + 8 + (6*n^3 - 11*n^2 + 15*n - 22)*x + (12*n^2 - 27*n + 18)*x^2 + (6*n - 4)*x^3)*P[n - 1, x] + ((4*n^2 - 4*n + 6)*x - (4*n^2 - 12*n + 16)*x^2 - (8*n - 14)*x^3 - 4*x^4)*D[P[n - 1, x], x])/(2*n^3 - 7*n^2 + 12*n - 7 + (4*n^2 - 13*n + 9)*x + (2*n - 2)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 2; for(n = 2, N, my(i = n); v[i] = ((2*n^3 + 3*n^2 + 2*n + 8 + (6*n^3 - 11*n^2 + 15*n - 22)*x + (12*n^2 - 27*n + 18)*x^2 + (6*n - 4)*x^3)*v[i - 1] + ((4*n^2 - 4*n + 6)*x - (4*n^2 - 12*n + 16)*x^2 - (8*n - 14)*x^3 - 4*x^4)*deriv(v[i - 1], x))/(2*n^3 - 7*n^2 + 12*n - 7 + (4*n^2 - 13*n + 9)*x + (2*n - 2)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.