Real-rooted atlas

A128099 → D → D+1

nonnegative coefficients

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

Recurrence

\(\left(-44 + 70\,x - 16\,x^{2} + 48\,n - 51\,n x + 8\,n x^{2} - 17\,n^{2} + 8\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(-22 - 193\,x + 316\,x^{2} - 96\,x^{3} + 29\,n + 189\,n x - 196\,n x^{2} + 32\,n x^{3} - 13\,n^{2} - 68\,n^{2} x + 32\,n^{2} x^{2} + 2\,n^{3} + 8\,n^{3} x\right)\,P_{n-1}(x) + \left(-11\,x - 69\,x^{2} + 144\,x^{3} - 64\,x^{4} + 9\,n x + 64\,n x^{2} - 64\,n x^{3} - 2\,n^{2} x - 16\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
10
222
344
414822
5322456
6589624178
79227296460
8134600432641230
9184112814403203072
102421904372016001607626
1130829768064608096018388
1238243921545617920528038443814
1346462002707243904224002688102728
14554844844280940807392016128896238306
Technical details
Node id
n-2eb44f0a3017f647 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-31b39615a99b215c

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half affine: slope 1, intercept -2)
  • 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:   0;
  2:   2;
  3:   4;
  4:  14,    8;
  5:  32,   24;
  6:  58,   96,   24;
  7:  92,  272,   96;
  8: 134,  600,  432,  64;
  9: 184, 1128, 1440, 320;
%F Formula
The row polynomials satisfy (2*n^3-17*n^2+48*n-44+(8*n^2-51*n+70)*x+(8*n-16)*x^2)*P(n,x) = (2*n^3-13*n^2+29*n-22+(8*n^3-68*n^2+189*n-193)*x+(32*n^2-196*n+316)*x^2+(32*n-96)*x^3)*P(n-1,x) - ((2*n^2-9*n+11)*x+(16*n^2-64*n+69)*x^2+(64*n-144)*x^3+64*x^4)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((2*n^3 - 13*n^2 + 29*n - 22 + (8*n^3 - 68*n^2 + 189*n - 193)*x + (32*n^2 - 196*n + 316)*x^2 + (32*n - 96)*x^3)*P[n - 1, x] - ((2*n^2 - 9*n + 11)*x + (16*n^2 - 64*n + 69)*x^2 + (64*n - 144)*x^3 + 64*x^4)*D[P[n - 1, x], x])/(2*n^3 - 17*n^2 + 48*n - 44 + (8*n^2 - 51*n + 70)*x + (8*n - 16)*x^2)]];
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] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = ((2*n^3 - 13*n^2 + 29*n - 22 + (8*n^3 - 68*n^2 + 189*n - 193)*x + (32*n^2 - 196*n + 316)*x^2 + (32*n - 96)*x^3)*v[i - 1] - ((2*n^2 - 9*n + 11)*x + (16*n^2 - 64*n + 69)*x^2 + (64*n - 144)*x^3 + 64*x^4)*deriv(v[i - 1], x))/(2*n^3 - 17*n^2 + 48*n - 44 + (8*n^2 - 51*n + 70)*x + (8*n - 16)*x^2)); 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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.