Real-rooted atlas

A026729 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(16 + 32\,x - 16\,n - 16\,n x + 4\,n^{2}\right)\,P_{n}(x) = \left(-120 - 256\,x - 128\,x^{2} + 68\,n + 80\,n x - 8\,n^{2}\right)\,P_{n-1}(x) + \left(-15 - 92\,x - 144\,x^{2} - 64\,x^{3} + 5\,n + 24\,n x + 16\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
244
3123244
428160192380
560512134410242940
6120137655689216512021400
7232332818240471045632024576149800
84367520520321832963430403194881146881020500

All 15 rows

Technical details
Node id
n-931ff6b0beecaef6 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-5ae4b7134ed59725

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:    4;
   3:   12,    32;
   4:   28,   160,    192;
   5:   60,   512,   1344,    1024;
   6:  120,  1376,   5568,    9216,    5120;
   7:  232,  3328,  18240,   47104,   56320,    24576;
   8:  436,  7520,  52032,  183296,  343040,   319488,   114688;
   9:  804, 16192, 135360,  605184, 1551360,  2260992,  1720320,  524288;
  10: 1460, 33632, 329664, 1787904, 5836800, 11673600, 13877248, 8912896, 2359296;
%F Formula
The row polynomials satisfy (4*n^2-16*n+16-(16*n-32)*x)*P(n,x) = -(8*n^2-68*n+120-(80*n-256)*x+128*x^2)*P(n-1,x) + (5*n-15+(24*n-92)*x+(16*n-144)*x^2-64*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^2 - 68*n + 120 - (80*n - 256)*x + 128*x^2)*P[n - 1, x] + (5*n - 15 + (24*n - 92)*x + (16*n - 144)*x^2 - 64*x^3)*D[P[n - 1, x], x])/(4*n^2 - 16*n + 16 - (16*n - 32)*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] = 4; for(n = 3, N, my(i = n); v[i] = (-(8*n^2 - 68*n + 120 - (80*n - 256)*x + 128*x^2)*v[i - 1] + (5*n - 15 + (24*n - 92)*x + (16*n - 144)*x^2 - 64*x^3)*deriv(v[i - 1], x))/(4*n^2 - 16*n + 16 - (16*n - 32)*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

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.