Real-rooted atlas

A062145 → D → D

nonnegative coefficients

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

Recurrence

\(\left(-3\,n + n^{2}\right)\,P_{n}(x) = \left(-2 + 4\,x + n - 7\,n x + n^{2} + 3\,n^{2} x\right)\,P_{n-1}(x) + \left(-x + x^{2} + 2\,n x - 2\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 0\)
\(P_{3}(x) = 20\)

Rows ?

nx0x1x2x3x4x5Pn(1)
10
20
32020
490120210
52528404201512
65603360420011209240
71080100802268015120252051480
818902520088200105840441005040270270

All 15 rows

Technical details
Node id
n-64747085fe504280 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-a63b17798047f45f

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 (affine: slope 1, intercept -3)
  • 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:    0;
   2:    0;
   3:   20;
   4:   90,    120;
   5:  252,    840,    420;
   6:  560,   3360,   4200,    1120;
   7: 1080,  10080,  22680,   15120,    2520;
   8: 1890,  25200,  88200,  105840,   44100,    5040;
   9: 3080,  55440, 277200,  517440,  388080,  110880,   9240;
  10: 4752, 110880, 748440, 1995840, 2328480, 1197504, 249480, 15840;
%F Formula
The row polynomials satisfy (n^2-3*n)*P(n,x) = (n^2+n-2+(3*n^2-7*n+4)*x)*P(n-1,x) + ((2*n-1)*x-(2*n-1)*x^2)*d/dx P(n-1,x) for n >= 4, with P(1,x) = 0, P(2,x) = 0, P(3,x) = 20.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 0;
P[3, x_] := 20;
P[n_, x_] := P[n, x] = Expand[Together[((n^2 + n - 2 + (3*n^2 - 7*n + 4)*x)*P[n - 1, x] + ((2*n - 1)*x - (2*n - 1)*x^2)*D[P[n - 1, x], x])/(n^2 - 3*n)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 0; v[3] = 20; for(n = 4, N, my(i = n); v[i] = ((n^2 + n - 2 + (3*n^2 - 7*n + 4)*x)*v[i - 1] + ((2*n - 1)*x - (2*n - 1)*x^2)*deriv(v[i - 1], x))/(n^2 - 3*n)); 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

Same fingerprint ?

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.