Real-rooted atlas

A026729 → basis A008277 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-x^{2}\right)\,P_{n-4}(x) + \left(-x^{2}\right)\,P_{n-4}'(x) + \left(-2\,x^{2}\right)\,P_{n-3}(x) + \left(-2\,x^{2}\right)\,P_{n-3}'(x) + \left(1 + 2\,x - x^{2}\right)\,P_{n-2}(x) + \left(x - x^{2}\right)\,P_{n-2}'(x) + \left(1 + 2\,x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + 2\,x\)
\(P_{3}(x) = 2 + 8\,x + 3\,x^{2}\)
\(P_{4}(x) = 3 + 26\,x + 24\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2123
328313
432624457
5578132525272
682246244329561389
71363027212944110515677576
82117501131617964102202412238843929
934482845681102348827602899847113449269713
105513266180858557244615290300036712888472477101746996
118936366706758294042043147402811864915866157384142926401111898430
12144995522737296151741442902223024587436105406282455024319509229108361284959721
13233272302105360397705032818932805520440016411216130833992400594316860639035222106813634326690
143777444624037778638659566812070320151635945942112606575243181017697367445132515301088868522961339144940333670
156102034748154266621192246908475613578451271770872010814659877514153724814538234152533687702760870218666487538716521540050779342
Technical details
Node id
n-9a641acc1d377232 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-9c3ef9fd36e933df

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,     2;
   3:  2,     8,      3;
   4:  3,    26,     24,      4;
   5:  5,    78,    132,     52,      5;
   6:  8,   224,    624,    432,     95,      6;
   7: 13,   630,   2721,   2944,   1105,    156,     7;
   8: 21,  1750,  11316,  17964,  10220,   2412,   238,    8;
   9: 34,  4828,  45681, 102348,  82760,  28998,  4711,  344,   9;
  10: 55, 13266, 180858, 557244, 615290, 300036, 71288, 8472, 477, 10;
%F Formula
The row polynomials satisfy P(n,x) = -x^2*P(n-4,x) - x^2*d/dx P(n-4,x) - 2*x^2*P(n-3,x) - 2*x^2*d/dx P(n-3,x) + (1+2*x-x^2)*P(n-2,x) + (x-x^2)*d/dx P(n-2,x) + (1+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 5, with P(1,x) = 1, P(2,x) = 1 + 2*x, P(3,x) = 2 + 8*x + 3*x^2, P(4,x) = 3 + 26*x + 24*x^2 + 4*x^3.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + 2*x;
P[3, x_] := 2 + 8*x + 3*x^2;
P[4, x_] := 3 + 26*x + 24*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[-x^2*P[n - 4, x] - x^2*D[P[n - 4, x], x] - 2*x^2*P[n - 3, x] - 2*x^2*D[P[n - 3, x], x] + (1 + 2*x - x^2)*P[n - 2, x] + (x - x^2)*D[P[n - 2, x], x] + (1 + 2*x)*P[n - 1, x] + x*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 + 2*x; v[3] = 2 + 8*x + 3*x^2; v[4] = 3 + 26*x + 24*x^2 + 4*x^3; for(n = 5, N, my(i = n); v[i] = -x^2*v[i - 4] - x^2*deriv(v[i - 4], x) - 2*x^2*v[i - 3] - 2*x^2*deriv(v[i - 3], x) + (1 + 2*x - x^2)*v[i - 2] + (x - x^2)*deriv(v[i - 2], x) + (1 + 2*x)*v[i - 1] + x*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.