Real-rooted atlas

A063967 → basis A013609 → D

nonnegative coefficients

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

Recurrence

\(\left(6 + 4\,x - 5\,n - 2\,n x + n^{2}\right)\,P_{n}(x) = \left(-48 - 40\,x - 8\,x^{2} + 22\,n + 10\,n x - 2\,n^{2}\right)\,P_{n-1}(x) + \left(-24 - 44\,x - 24\,x^{2} - 4\,x^{3} + 6\,n + 8\,n x + 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
222
310818
4406424128
514434426464816
6488155218008961604896
71592632098167424272038428256
85056240644684847616265607680896158720

All 15 rows

Technical details
Node id
n-42fb76becd83dd16 · 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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     0;
   2:     2;
   3:    10,      8;
   4:    40,     64,     24;
   5:   144,    344,    264,      64;
   6:   488,   1552,   1800,     896,     160;
   7:  1592,   6320,   9816,    7424,    2720,    384;
   8:  5056,  24064,  46848,   47616,   26560,   7680,    896;
   9: 15744,  87360, 204480,  261184,  196160,  86400,  20608,  2048;
  10: 48288, 306048, 836928, 1287808, 1217440, 722688, 262528, 53248, 4608;
%F Formula
The row polynomials satisfy (n^2-5*n+6-(2*n-4)*x)*P(n,x) = -(2*n^2-22*n+48-(10*n-40)*x+8*x^2)*P(n-1,x) + (6*n-24+(8*n-44)*x+(2*n-24)*x^2-4*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n^2 - 22*n + 48 - (10*n - 40)*x + 8*x^2)*P[n - 1, x] + (6*n - 24 + (8*n - 44)*x + (2*n - 24)*x^2 - 4*x^3)*D[P[n - 1, x], x])/(n^2 - 5*n + 6 - (2*n - 4)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2; for(n = 3, N, my(i = n); v[i] = (-(2*n^2 - 22*n + 48 - (10*n - 40)*x + 8*x^2)*v[i - 1] + (6*n - 24 + (8*n - 44)*x + (2*n - 24)*x^2 - 4*x^3)*deriv(v[i - 1], x))/(n^2 - 5*n + 6 - (2*n - 4)*x)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.