Real-rooted atlas

A145596 → D → D

nonnegative coefficients

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

Recurrence

\(\left(-8\,n + 2\,n^{2} + n^{3}\right)\,P_{n}(x) = \left(8 - 6\,x + 14\,n - 6\,n x + 7\,n^{2} + 3\,n^{2} x + n^{3} + 3\,n^{3} x\right)\,P_{n-1}(x) + \left(2\,x - 2\,x^{2} + 4\,n x - 4\,n x^{2} + 2\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 6\)

Rows ?

nx0x1x2x3x4x5Pn(1)
00
10
266
3402464
415024060450
542012608401202640
698047045880224021014014
72016141122822420160504033669888

All 15 rows

Technical details
Node id
n-96653488893d491a · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-82e79442740d0b27

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 -2)
  • 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:    0;
  1:    0;
  2:    6;
  3:   40,    24;
  4:  150,   240,     60;
  5:  420,  1260,    840,    120;
  6:  980,  4704,   5880,   2240,    210;
  7: 2016, 14112,  28224,  20160,   5040,    336;
  8: 3780, 36288, 105840, 120960,  56700,  10080,   504;
  9: 6600, 83160, 332640, 554400, 415800, 138600, 18480, 720;
%F Formula
The row polynomials satisfy (n^3+2*n^2-8*n)*P(n,x) = (n^3+7*n^2+14*n+8+(3*n^3+3*n^2-6*n-6)*x)*P(n-1,x) + ((2*n^2+4*n+2)*x-(2*n^2+4*n+2)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 6.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 6;
P[n_, x_] := P[n, x] = Expand[Together[((n^3 + 7*n^2 + 14*n + 8 + (3*n^3 + 3*n^2 - 6*n - 6)*x)*P[n - 1, x] + ((2*n^2 + 4*n + 2)*x - (2*n^2 + 4*n + 2)*x^2)*D[P[n - 1, x], x])/(n^3 + 2*n^2 - 8*n)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 0; v[3] = 6; for(n = 3, N, my(i = n + 1); v[i] = ((n^3 + 7*n^2 + 14*n + 8 + (3*n^3 + 3*n^2 - 6*n - 6)*x)*v[i - 1] + ((2*n^2 + 4*n + 2)*x - (2*n^2 + 4*n + 2)*x^2)*deriv(v[i - 1], x))/(n^3 + 2*n^2 - 8*n)); 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.