Real-rooted atlas

A156289 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-4 - 10\,x - 15\,x^{2}\right)\,P_{n-2}(x) + \left(-5\,x - 14\,x^{2} - 6\,x^{3}\right)\,P_{n-2}'(x) + \left(-x^{2} - 3\,x^{3}\right)\,P_{n-2}''(x) + \left(5 + 10\,x\right)\,P_{n-1}(x) + \left(5\,x + 2\,x^{2}\right)\,P_{n-1}'(x) + \left(x^{2}\right)\,P_{n-1}''(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 3\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
233
3153045
463420315798
525544109450378017895
610234224019750520790051975500643
7409539039036036007567560472972581081017106180
81638335544606175669523243220027432405011351340014189175699786363

All 15 rows

Technical details
Node id
n-9212bcd5fb1e697f · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-48c328a3a13ffc6f

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 -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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:      0;
   2:      3;
   3:     15,        30;
   4:     63,       420,         315;
   5:    255,      4410,        9450,         3780;
   6:   1023,     42240,      197505,       207900,        51975;
   7:   4095,    390390,     3603600,      7567560,      4729725,       810810;
   8:  16383,   3554460,    61756695,    232432200,    274324050,    113513400,     14189175;
   9:  65535,  32153970,  1025428950,   6551885340,  13071608550,   9841611780,   2894591700,   275675400;
  10: 262143, 290041080, 16744479885, 176102280900, 560666331225, 686156070600, 357482074950, 78567489000, 5892561675;
%F Formula
The row polynomials satisfy P(n,x) = -(4+10*x+15*x^2)*P(n-2,x) - (5*x+14*x^2+6*x^3)*d/dx P(n-2,x) - (x^2+3*x^3)*d^2/dx^2 P(n-2,x) + (5+10*x)*P(n-1,x) + (5*x+2*x^2)*d/dx P(n-1,x) + x^2*d^2/dx^2 P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 3.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 3;
P[n_, x_] := P[n, x] = Expand[-(4 + 10*x + 15*x^2)*P[n - 2, x] - (5*x + 14*x^2 + 6*x^3)*D[P[n - 2, x], x] - (x^2 + 3*x^3)*D[P[n - 2, x], {x, 2}] + (5 + 10*x)*P[n - 1, x] + (5*x + 2*x^2)*D[P[n - 1, x], x] + x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 3; for(n = 3, N, my(i = n); v[i] = -(4 + 10*x + 15*x^2)*v[i - 2] - (5*x + 14*x^2 + 6*x^3)*deriv(v[i - 2], x) - (x^2 + 3*x^3)*deriv(deriv(v[i - 2], x), x) + (5 + 10*x)*v[i - 1] + (5*x + 2*x^2)*deriv(v[i - 1], x) + x^2*deriv(deriv(v[i - 1], x), 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.