Real-rooted atlas

A026729 → basis A008277 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4\,x + 9\,x^{2}\right)\,P_{n-4}(x) + \left(-2\,x + 4\,x^{2} + 4\,x^{3}\right)\,P_{n-4}'(x) + \left(-x^{2} + 4\,x^{3}\right)\,P_{n-4}''(x) + \left(-8\,x + 18\,x^{2}\right)\,P_{n-3}(x) + \left(-4\,x + 8\,x^{2} + 8\,x^{3}\right)\,P_{n-3}'(x) + \left(-2\,x^{2} + 8\,x^{3}\right)\,P_{n-3}''(x) + \left(1 + 9\,x^{2}\right)\,P_{n-2}(x) + \left(4\,x^{3}\right)\,P_{n-2}'(x) + \left(-x^{2} + 4\,x^{3}\right)\,P_{n-2}''(x) + \left(1 + 4\,x\right)\,P_{n-1}(x) + \left(2\,x - 4\,x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 4\,x\)
\(P_{3}(x) = 2 + 16\,x + 9\,x^{2}\)
\(P_{4}(x) = 3 + 52\,x + 72\,x^{2} + 16\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2145
3216927
43527216143
5515639620825790
6844818721728475364567
713126081631177655259364927722
821350033948718565110014472166664176627

All 15 rows

Technical details
Node id
n-3717f46990c397a2 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-822bc1e7f59d7f29

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:  1,     4;
   3:  2,    16,      9;
   4:  3,    52,     72,      16;
   5:  5,   156,    396,     208,      25;
   6:  8,   448,   1872,    1728,     475,      36;
   7: 13,  1260,   8163,   11776,    5525,     936,     49;
   8: 21,  3500,  33948,   71856,   51100,   14472,   1666,    64;
   9: 34,  9656, 137043,  409392,  413800,  173988,  32977,  2752,   81;
  10: 55, 26532, 542574, 2228976, 3076450, 1800216, 499016, 67776, 4293, 100;
%F Formula
The row polynomials satisfy P(n,x) = -(4*x-9*x^2)*P(n-4,x) - (2*x-4*x^2-4*x^3)*d/dx P(n-4,x) - (x^2-4*x^3)*d^2/dx^2 P(n-4,x) - (8*x-18*x^2)*P(n-3,x) - (4*x-8*x^2-8*x^3)*d/dx P(n-3,x) - (2*x^2-8*x^3)*d^2/dx^2 P(n-3,x) + (1+9*x^2)*P(n-2,x) + 4*x^3*d/dx P(n-2,x) - (x^2-4*x^3)*d^2/dx^2 P(n-2,x) + (1+4*x)*P(n-1,x) + (2*x-4*x^2)*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1 + 4*x, P(3,x) = 2 + 16*x + 9*x^2, P(4,x) = 3 + 52*x + 72*x^2 + 16*x^3.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 4*x;
P[3, x_] := 2 + 16*x + 9*x^2;
P[4, x_] := 3 + 52*x + 72*x^2 + 16*x^3;
P[n_, x_] := P[n, x] = Expand[-(4*x - 9*x^2)*P[n - 4, x] - (2*x - 4*x^2 - 4*x^3)*D[P[n - 4, x], x] - (x^2 - 4*x^3)*D[P[n - 4, x], {x, 2}] - (8*x - 18*x^2)*P[n - 3, x] - (4*x - 8*x^2 - 8*x^3)*D[P[n - 3, x], x] - (2*x^2 - 8*x^3)*D[P[n - 3, x], {x, 2}] + (1 + 9*x^2)*P[n - 2, x] + 4*x^3*D[P[n - 2, x], x] - (x^2 - 4*x^3)*D[P[n - 2, x], {x, 2}] + (1 + 4*x)*P[n - 1, x] + (2*x - 4*x^2)*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + 4*x; v[3] = 2 + 16*x + 9*x^2; v[4] = 3 + 52*x + 72*x^2 + 16*x^3; for(n = 5, N, my(i = n); v[i] = -(4*x - 9*x^2)*v[i - 4] - (2*x - 4*x^2 - 4*x^3)*deriv(v[i - 4], x) - (x^2 - 4*x^3)*deriv(deriv(v[i - 4], x), x) - (8*x - 18*x^2)*v[i - 3] - (4*x - 8*x^2 - 8*x^3)*deriv(v[i - 3], x) - (2*x^2 - 8*x^3)*deriv(deriv(v[i - 3], x), x) + (1 + 9*x^2)*v[i - 2] + 4*x^3*deriv(v[i - 2], x) - (x^2 - 4*x^3)*deriv(deriv(v[i - 2], x), x) + (1 + 4*x)*v[i - 1] + (2*x - 4*x^2)*deriv(v[i - 1], 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. [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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.