Real-rooted atlas

A099040 → basis A007318 → D

nonnegative coefficients

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

Recurrence

\(\left(2 + 2\,x - 3\,n - 2\,n x + n^{2}\right)\,P_{n}(x) = \left(-28 - 30\,x - 8\,x^{2} + 18\,n + 10\,n x - 2\,n^{2}\right)\,P_{n-1}(x) + \left(-18 - 36\,x - 22\,x^{2} - 4\,x^{3} + 6\,n + 8\,n x + 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
210818
3406424128
414434426464816
5488155218008961604896
61592632098167424272038428256
75056240644684847616265607680896158720

All 15 rows

Technical details
Node id
n-6a259e6ccb673e65 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-836bab44144e8751

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:    10,      8;
  3:    40,     64,     24;
  4:   144,    344,    264,      64;
  5:   488,   1552,   1800,     896,     160;
  6:  1592,   6320,   9816,    7424,    2720,    384;
  7:  5056,  24064,  46848,   47616,   26560,   7680,    896;
  8: 15744,  87360, 204480,  261184,  196160,  86400,  20608,  2048;
  9: 48288, 306048, 836928, 1287808, 1217440, 722688, 262528, 53248, 4608;
%F Formula
The row polynomials satisfy (n^2-3*n+2-(2*n-2)*x)*P(n,x) = -(2*n^2-18*n+28-(10*n-30)*x+8*x^2)*P(n-1,x) + (6*n-18+(8*n-36)*x+(2*n-22)*x^2-4*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[(-(2*n^2 - 18*n + 28 - (10*n - 30)*x + 8*x^2)*P[n - 1, x] + (6*n - 18 + (8*n - 36)*x + (2*n - 22)*x^2 - 4*x^3)*D[P[n - 1, x], x])/(n^2 - 3*n + 2 - (2*n - 2)*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] = (-(2*n^2 - 18*n + 28 - (10*n - 30)*x + 8*x^2)*v[i - 1] + (6*n - 18 + (8*n - 36)*x + (2*n - 22)*x^2 - 4*x^3)*deriv(v[i - 1], x))/(n^2 - 3*n + 2 - (2*n - 2)*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

Same fingerprint ?

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.