Real-rooted atlas

A035607 → basis A007318 → D+1

nonnegative coefficients

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

Recurrence

\(\left(8 + 6\,x - 14\,x^{2} - 14\,x^{3} - 2\,x^{4} - 8\,n - 24\,n x - 16\,n x^{2} + 4\,n x^{3} + n x^{4} + 2\,n^{2} x + 8\,n^{2} x^{2} + n^{2} x^{3} + 2\,n^{3} x\right)\,P_{n}(x) = \left(-8 - 6\,x + 14\,x^{2} + 14\,x^{3} + 2\,x^{4} + 16\,n + 22\,n x - 12\,n x^{2} - 20\,n x^{3} - 3\,n x^{4} - 24\,n^{2} x - 24\,n^{2} x^{2} - 3\,n^{2} x^{3} - 8\,n^{3} - 6\,n^{3} x\right)\,P_{n-1}(x) + \left(32 + 152\,x + 276\,x^{2} + 239\,x^{3} + 99\,x^{4} + 17\,x^{5} + x^{6} + 48\,n + 160\,n x + 190\,n x^{2} + 94\,n x^{3} + 17\,n x^{4} + n x^{5} + 16\,n^{2} + 32\,n^{2} x + 18\,n^{2} x^{2} + 2\,n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
011
111
2527
31512229
44556202123
5129220126282505
63617866282263622039
79912632271613763564428117

All 15 rows

Technical details
Node id
n-96da891f206ebda7 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-b074c357d42fbaf1

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (irregular)
  • 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:    1;
  2:    5,     2;
  3:   15,    12,     2;
  4:   45,    56,    20,     2;
  5:  129,   220,   126,    28,     2;
  6:  361,   786,   628,   226,    36,    2;
  7:  991,  2632,  2716,  1376,   356,   44,   2;
  8: 2681,  8416, 10680,  7036,  2572,  516,  52,  2;
  9: 7169, 25992, 39220, 31944, 15240, 4324, 706, 60, 2;
%F Formula
The row polynomials satisfy -(8*n-8-(2*n^3+2*n^2-24*n+6)*x-(8*n^2-16*n-14)*x^2-(n^2+4*n-14)*x^3-(n-2)*x^4)*P(n,x) = -(8*n^3-16*n+8+(6*n^3+24*n^2-22*n+6)*x+(24*n^2+12*n-14)*x^2+(3*n^2+20*n-14)*x^3+(3*n-2)*x^4)*P(n-1,x) + (16*n^2+48*n+32+(32*n^2+160*n+152)*x+(18*n^2+190*n+276)*x^2+(2*n^2+94*n+239)*x^3+(17*n+99)*x^4+(n+17)*x^5+x^6)*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[(-(8*n^3 - 16*n + 8 + (6*n^3 + 24*n^2 - 22*n + 6)*x + (24*n^2 + 12*n - 14)*x^2 + (3*n^2 + 20*n - 14)*x^3 + (3*n - 2)*x^4)*P[n - 1, x] + (16*n^2 + 48*n + 32 + (32*n^2 + 160*n + 152)*x + (18*n^2 + 190*n + 276)*x^2 + (2*n^2 + 94*n + 239)*x^3 + (17*n + 99)*x^4 + (n + 17)*x^5 + x^6)*D[P[n - 1, x], x])/(-8*n + 8 + (2*n^3 + 2*n^2 - 24*n + 6)*x + (8*n^2 - 16*n - 14)*x^2 + (n^2 + 4*n - 14)*x^3 + (n - 2)*x^4)]];
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] = (-(8*n^3 - 16*n + 8 + (6*n^3 + 24*n^2 - 22*n + 6)*x + (24*n^2 + 12*n - 14)*x^2 + (3*n^2 + 20*n - 14)*x^3 + (3*n - 2)*x^4)*v[i - 1] + (16*n^2 + 48*n + 32 + (32*n^2 + 160*n + 152)*x + (18*n^2 + 190*n + 276)*x^2 + (2*n^2 + 94*n + 239)*x^3 + (17*n + 99)*x^4 + (n + 17)*x^5 + x^6)*deriv(v[i - 1], x))/(-8*n + 8 + (2*n^3 + 2*n^2 - 24*n + 6)*x + (8*n^2 - 16*n - 14)*x^2 + (n^2 + 4*n - 14)*x^3 + (n - 2)*x^4)); 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.