Real-rooted atlas

A119900 → D+1

nonnegative coefficients

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

Recurrence

\(\left(8\,n x - 14\,n x^{2} + 4\,n x^{3} + n^{2} - 10\,n^{2} x + 8\,n^{2} x^{2} - n^{3} + 4\,n^{3} x\right)\,P_{n}(x) = \left(6\,x - 16\,x^{2} + 14\,x^{3} - 4\,x^{4} + 3\,n - 13\,n x + 30\,n x^{2} - 42\,n x^{3} + 12\,n x^{4} - 4\,n^{2} + 18\,n^{2} x - 34\,n^{2} x^{2} + 24\,n^{2} x^{3} + n^{3} - 7\,n^{3} x + 12\,n^{3} x^{2}\right)\,P_{n-1}(x) + \left(4\,x^{2} - 16\,x^{3} + 20\,x^{4} - 8\,x^{5} + 2\,n x - 10\,n x^{2} + 24\,n x^{3} - 16\,n x^{4} - 2\,n^{2} x + 10\,n^{2} x^{2} - 8\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1224
217311
30816428
4023130568
5001886506160
600385196777368
7000322883921128832

All 15 rows

Technical details
Node id
n-8be6d20222793ca7 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-06cde147478dd591

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 0)
  • 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: 1;
  1: 2, 2;
  2: 1, 7,  3;
  3: 0, 8, 16,  4;
  4: 0, 2, 31, 30,   5;
  5: 0, 0, 18, 86,  50,   6;
  6: 0, 0,  3, 85, 196,  77,    7;
  7: 0, 0,  0, 32, 288, 392,  112,    8;
  8: 0, 0,  0,  4, 181, 792,  714,  156,   9;
  9: 0, 0,  0,  0,  50, 730, 1884, 1212, 210, 10;
%F Formula
The row polynomials satisfy -(n^3-n^2-(4*n^3-10*n^2+8*n)*x-(8*n^2-14*n)*x^2-4*n*x^3)*P(n,x) = (n^3-4*n^2+3*n-(7*n^3-18*n^2+13*n-6)*x+(12*n^3-34*n^2+30*n-16)*x^2+(24*n^2-42*n+14)*x^3+(12*n-4)*x^4)*P(n-1,x) - ((2*n^2-2*n)*x-(10*n^2-10*n+4)*x^2+(8*n^2-24*n+16)*x^3+(16*n-20)*x^4+8*x^5)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 - 4*n^2 + 3*n - (7*n^3 - 18*n^2 + 13*n - 6)*x + (12*n^3 - 34*n^2 + 30*n - 16)*x^2 + (24*n^2 - 42*n + 14)*x^3 + (12*n - 4)*x^4)*P[n - 1, x] - ((2*n^2 - 2*n)*x - (10*n^2 - 10*n + 4)*x^2 + (8*n^2 - 24*n + 16)*x^3 + (16*n - 20)*x^4 + 8*x^5)*D[P[n - 1, x], x])/(-n^3 + n^2 + (4*n^3 - 10*n^2 + 8*n)*x + (8*n^2 - 14*n)*x^2 + 4*n*x^3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n^3 - 4*n^2 + 3*n - (7*n^3 - 18*n^2 + 13*n - 6)*x + (12*n^3 - 34*n^2 + 30*n - 16)*x^2 + (24*n^2 - 42*n + 14)*x^3 + (12*n - 4)*x^4)*v[i - 1] - ((2*n^2 - 2*n)*x - (10*n^2 - 10*n + 4)*x^2 + (8*n^2 - 24*n + 16)*x^3 + (16*n - 20)*x^4 + 8*x^5)*deriv(v[i - 1], x))/(-n^3 + n^2 + (4*n^3 - 10*n^2 + 8*n)*x + (8*n^2 - 14*n)*x^2 + 4*n*x^3)); 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.