Real-rooted atlas

A225117 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-5 + 3\,n\right)\,P_{n}(x) = \left(-4 + 32\,x + 80\,x^{2} + 6\,n - 60\,n x - 198\,n x^{2} + 18\,n^{2} x + 135\,n^{2} x^{2} - 27\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-6\,x - 24\,x^{2} + 30\,x^{3} + 9\,n x + 54\,n x^{2} - 63\,n x^{3} - 27\,n^{2} x^{2} + 27\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-8 + 50\,x + 3\,n - 60\,n x + 18\,n^{2} x\right)\,P_{n-1}(x) + \left(-15\,x + 15\,x^{2} + 9\,n x - 9\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2224
3426333
481861804378
51610903393100455508
632586643494427125090697200
7643035446233910497524449902453472012040
8128154202442720819561148193517604154994114632847764080

All 15 rows

Technical details
Node id
n-fd5a252118896eef · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e5c1d65b9edad3e4

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:   1;
   2:   2,       2;
   3:   4,      26,         3;
   4:   8,     186,       180,          4;
   5:  16,    1090,      3393,       1004,           5;
   6:  32,    5866,     43494,      42712,        5090,           6;
   7:  64,   30354,    462339,    1049752,      444990,       24534,          7;
   8: 128,  154202,   4427208,   19561148,    19351760,     4154994,     114632,         8;
   9: 256,  776642,  39812421,  309619732,   588793295,   302856018,   36225091,    524216,       9;
  10: 512, 3896010, 344128554, 4414398384, 14435500770, 14333727708, 4257822030, 301857696, 2359206, 10;
%F Formula
The row polynomials satisfy (3*n-5)*P(n,x) = (6*n-4+(18*n^2-60*n+32)*x-(27*n^3-135*n^2+198*n-80)*x^2)*P(n-2,x) + ((9*n-6)*x-(27*n^2-54*n+24)*x^2+(27*n^2-63*n+30)*x^3)*d/dx P(n-2,x) + (3*n-8+(18*n^2-60*n+50)*x)*P(n-1,x) + ((9*n-15)*x-(9*n-15)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2 + 2*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[((6*n - 4 + (18*n^2 - 60*n + 32)*x - (27*n^3 - 135*n^2 + 198*n - 80)*x^2)*P[n - 2, x] + ((9*n - 6)*x - (27*n^2 - 54*n + 24)*x^2 + (27*n^2 - 63*n + 30)*x^3)*D[P[n - 2, x], x] + (3*n - 8 + (18*n^2 - 60*n + 50)*x)*P[n - 1, x] + ((9*n - 15)*x - (9*n - 15)*x^2)*D[P[n - 1, x], x])/(3*n - 5)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2 + 2*x; for(n = 3, N, my(i = n); v[i] = ((6*n - 4 + (18*n^2 - 60*n + 32)*x - (27*n^3 - 135*n^2 + 198*n - 80)*x^2)*v[i - 2] + ((9*n - 6)*x - (27*n^2 - 54*n + 24)*x^2 + (27*n^2 - 63*n + 30)*x^3)*deriv(v[i - 2], x) + (3*n - 8 + (18*n^2 - 60*n + 50)*x)*v[i - 1] + ((9*n - 15)*x - (9*n - 15)*x^2)*deriv(v[i - 1], x))/(3*n - 5)); 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. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.