Real-rooted atlas

A269939 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-3 - 2\,x + n\right)\,P_{n}(x) = \left(8\,x^{2} + 12\,x^{3} + 4\,x^{4} - 12\,n x^{2} - 12\,n x^{3} - 2\,n x^{4} + 6\,n^{2} x^{2} + 3\,n^{2} x^{3} - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-6\,x^{2} - 14\,x^{3} - 10\,x^{4} - 2\,x^{5} + 5\,n x^{2} + 8\,n x^{3} + 3\,n x^{4} - n^{2} x^{2} - n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(-3 + x + 6\,x^{2} + 2\,x^{3} + n - 7\,n x - 5\,n x^{2} + 2\,n^{2} x\right)\,P_{n-1}(x) + \left(-3\,x - 5\,x^{2} - 2\,x^{3} + n x + n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2112
317311
4121551592
51513405251051022
6111315265530598594514100
71239587339718960757969510395231996
81493207212347451009855181219512162151351354429360

All 15 rows

Technical details
Node id
n-47116cccf73aaef7 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-22d206e06a8a7627

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,    1;
   3: 1,    7,      3;
   4: 1,   21,     55,      15;
   5: 1,   51,    340,     525,      105;
   6: 1,  113,   1526,    5530,     5985,       945;
   7: 1,  239,   5873,   39718,    96075,     79695,     10395;
   8: 1,  493,  20721,  234745,  1009855,   1812195,   1216215,    135135;
   9: 1, 1003,  69306, 1234915,  8486170,  26231205,  37207170,  20945925,   2027025;
  10: 1, 2025, 223960, 6023116, 62097750, 297166870, 709668960, 829728900, 402026625, 34459425;
%F Formula
The row polynomials satisfy (n-3-2*x)*P(n,x) = -((n^3-6*n^2+12*n-8)*x^2-(3*n^2-12*n+12)*x^3+(2*n-4)*x^4)*P(n-2,x) - ((n^2-5*n+6)*x^2+(n^2-8*n+14)*x^3-(3*n-10)*x^4+2*x^5)*d/dx P(n-2,x) + (n-3+(2*n^2-7*n+1)*x-(5*n-6)*x^2+2*x^3)*P(n-1,x) + ((n-3)*x+(n-5)*x^2-2*x^3)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[Together[(-((n^3 - 6*n^2 + 12*n - 8)*x^2 - (3*n^2 - 12*n + 12)*x^3 + (2*n - 4)*x^4)*P[n - 2, x] - ((n^2 - 5*n + 6)*x^2 + (n^2 - 8*n + 14)*x^3 - (3*n - 10)*x^4 + 2*x^5)*D[P[n - 2, x], x] + (n - 3 + (2*n^2 - 7*n + 1)*x - (5*n - 6)*x^2 + 2*x^3)*P[n - 1, x] + ((n - 3)*x + (n - 5)*x^2 - 2*x^3)*D[P[n - 1, x], x])/(n - 3 - 2*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 + x; for(n = 3, N, my(i = n); v[i] = (-((n^3 - 6*n^2 + 12*n - 8)*x^2 - (3*n^2 - 12*n + 12)*x^3 + (2*n - 4)*x^4)*v[i - 2] - ((n^2 - 5*n + 6)*x^2 + (n^2 - 8*n + 14)*x^3 - (3*n - 10)*x^4 + 2*x^5)*deriv(v[i - 2], x) + (n - 3 + (2*n^2 - 7*n + 1)*x - (5*n - 6)*x^2 + 2*x^3)*v[i - 1] + ((n - 3)*x + (n - 5)*x^2 - 2*x^3)*deriv(v[i - 1], x))/(n - 3 - 2*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. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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.