Real-rooted atlas

A081581 → γ → θ+1

nonnegative coefficients

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

Recurrence

\(\left(n + 6\,n^{2} x\right)\,P_{n}(x) = \left(-24\,x - 144\,x^{2} + n + 24\,n x + 72\,n x^{2} + 6\,n^{2} x + 72\,n^{2} x^{2}\right)\,P_{n-1}(x) + \left(-x - 36\,x^{2} - 288\,x^{3} - 6\,n x^{2} - 144\,n x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3Pn(1)
011
111
211213
312425
4136108145
5148324373
61606488641573
7172108034564609

All 15 rows

Technical details
Node id
n-2106b6e2a6d1b702 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-71f62b7c405c9ec6

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half affine: slope 1, intercept 0)
  • 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: 1;
  1: 1;
  2: 1, 12;
  3: 1, 24;
  4: 1, 36,  108;
  5: 1, 48,  324;
  6: 1, 60,  648,   864;
  7: 1, 72, 1080,  3456;
  8: 1, 84, 1620,  8640,  6480;
  9: 1, 96, 2268, 17280, 32400;
%F Formula
The row polynomials satisfy (n+6*n^2*x)*P(n,x) = (n+(6*n^2+24*n-24)*x+(72*n^2+72*n-144)*x^2)*P(n-1,x) - (x+(6*n+36)*x^2+(144*n+288)*x^3)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((n + (6*n^2 + 24*n - 24)*x + (72*n^2 + 72*n - 144)*x^2)*P[n - 1, x] - (x + (6*n + 36)*x^2 + (144*n + 288)*x^3)*D[P[n - 1, x], x])/(n + 6*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = ((n + (6*n^2 + 24*n - 24)*x + (72*n^2 + 72*n - 144)*x^2)*v[i - 1] - (x + (6*n + 36)*x^2 + (144*n + 288)*x^3)*deriv(v[i - 1], x))/(n + 6*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

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.