Real-rooted atlas

A030528 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(8 + 8\,x - 2\,n - 8\,n^{2} - 8\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(40 + 16\,x - 116\,n - 144\,n x - 48\,n x^{2} + 44\,n^{2} + 32\,n^{2} x - 4\,n^{3}\right)\,P_{n-1}(x) + \left(10 + 24\,x + 8\,x^{2} - 25\,n - 80\,n x - 68\,n x^{2} - 16\,n x^{3} + 5\,n^{2} + 12\,n^{2} x + 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
00
122
28816
320482492
44416819264468
5904728646401602226
6176117629763520192038410152
7334271287841440012480537689644982
8620592823424492805920040320143362048195156
911321245658128149568230880217728121856368644608833220
1020402539213670441632078720095424073830435020892160102403512808
11363850536308376108544024328003615360359475223531529676802252802252814659542
1264329864067286426886406974400122976001499545612607488714240025907205406724915260664464
13112901894561428768639072018842400384902405572044857452544417715202083840067584001277952106496249278634
141969635898429659201468371248515200112748160189220864231696384206548992132126720588431361725235229818882293761018191384
Technical details
Node id
n-3e5bf5ad950f2219 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-34c2e490e5ac51c1

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:    2;
  2:    8,     8;
  3:   20,    48,    24;
  4:   44,   168,   192,     64;
  5:   90,   472,   864,    640,    160;
  6:  176,  1176,  2976,   3520,   1920,    384;
  7:  334,  2712,  8784,  14400,  12480,   5376,    896;
  8:  620,  5928, 23424,  49280,  59200,  40320,  14336,  2048;
  9: 1132, 12456, 58128, 149568, 230880, 217728, 121856, 36864, 4608;
%F Formula
The row polynomials satisfy (2*n^3-8*n^2-2*n+8-(8*n^2-8)*x)*P(n,x) = -(4*n^3-44*n^2+116*n-40-(32*n^2-144*n+16)*x+48*n*x^2)*P(n-1,x) + (5*n^2-25*n+10+(12*n^2-80*n+24)*x+(4*n^2-68*n+8)*x^2-16*n*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 44*n^2 + 116*n - 40 - (32*n^2 - 144*n + 16)*x + 48*n*x^2)*P[n - 1, x] + (5*n^2 - 25*n + 10 + (12*n^2 - 80*n + 24)*x + (4*n^2 - 68*n + 8)*x^2 - 16*n*x^3)*D[P[n - 1, x], x])/(2*n^3 - 8*n^2 - 2*n + 8 - (8*n^2 - 8)*x)]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 44*n^2 + 116*n - 40 - (32*n^2 - 144*n + 16)*x + 48*n*x^2)*v[i - 1] + (5*n^2 - 25*n + 10 + (12*n^2 - 80*n + 24)*x + (4*n^2 - 68*n + 8)*x^2 - 16*n*x^3)*deriv(v[i - 1], x))/(2*n^3 - 8*n^2 - 2*n + 8 - (8*n^2 - 8)*x)); 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.