Real-rooted atlas

A039756 → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(\left(-3 + 2\,n\right)\,P_{n}(x) = \left(6 + 20\,x + 15\,x^{2} - 24\,n - 68\,n x - 46\,n x^{2} + 28\,n^{2} + 64\,n^{2} x + 36\,n^{2} x^{2} - 8\,n^{3} - 16\,n^{3} x - 8\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(4 + 14\,x + 16\,x^{2} + 6\,x^{3} - 12\,n - 40\,n x - 44\,n x^{2} - 16\,n x^{3} + 8\,n^{2} + 24\,n^{2} x + 24\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(14 + 18\,x - 20\,n - 24\,n x + 8\,n^{2} + 8\,n^{2} x\right)\,P_{n-1}(x) + \left(6 + 12\,x + 6\,x^{2} - 4\,n - 8\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
111
2628
33832373
42563681324760
518443792247250458617
61429638472367921442418506105840
711916039817650641229974478990660071409089

All 15 rows

Technical details
Node id
n-957a5bf3e47ae72c · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-a8fa044417626521

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        0;
  1:        1;
  2:        6,        2;
  3:       38,       32,        3;
  4:      256,      368,      132,        4;
  5:     1844,     3792,     2472,      504,        5;
  6:    14296,    38472,    36792,    14424,     1850,        6;
  7:   119160,   398176,   506412,   299744,    78990,     6600,       7;
  8:  1063168,  4257920,  6797088,  5445024,  2240800,   421056,   23016,     8;
  9: 10107152, 47285888, 91141824, 92654272, 52564360, 16020192, 2225888, 78800, 9;
%F Formula
The row polynomials satisfy (2*n-3)*P(n,x) = -(8*n^3-28*n^2+24*n-6+(16*n^3-64*n^2+68*n-20)*x+(8*n^3-36*n^2+46*n-15)*x^2)*P(n-2,x) + (8*n^2-12*n+4+(24*n^2-40*n+14)*x+(24*n^2-44*n+16)*x^2+(8*n^2-16*n+6)*x^3)*d/dx P(n-2,x) + (8*n^2-20*n+14+(8*n^2-24*n+18)*x)*P(n-1,x) - (4*n-6+(8*n-12)*x+(4*n-6)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(8*n^3 - 28*n^2 + 24*n - 6 + (16*n^3 - 64*n^2 + 68*n - 20)*x + (8*n^3 - 36*n^2 + 46*n - 15)*x^2)*P[n - 2, x] + (8*n^2 - 12*n + 4 + (24*n^2 - 40*n + 14)*x + (24*n^2 - 44*n + 16)*x^2 + (8*n^2 - 16*n + 6)*x^3)*D[P[n - 2, x], x] + (8*n^2 - 20*n + 14 + (8*n^2 - 24*n + 18)*x)*P[n - 1, x] - (4*n - 6 + (8*n - 12)*x + (4*n - 6)*x^2)*D[P[n - 1, x], x])/(2*n - 3)]];
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] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(8*n^3 - 28*n^2 + 24*n - 6 + (16*n^3 - 64*n^2 + 68*n - 20)*x + (8*n^3 - 36*n^2 + 46*n - 15)*x^2)*v[i - 2] + (8*n^2 - 12*n + 4 + (24*n^2 - 40*n + 14)*x + (24*n^2 - 44*n + 16)*x^2 + (8*n^2 - 16*n + 6)*x^3)*deriv(v[i - 2], x) + (8*n^2 - 20*n + 14 + (8*n^2 - 24*n + 18)*x)*v[i - 1] - (4*n - 6 + (8*n - 12)*x + (4*n - 6)*x^2)*deriv(v[i - 1], x))/(2*n - 3)); 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. [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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.