Real-rooted atlas

A191935 → D

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
28816
32010424148
440584960641648
570218411472774416021630
611263847699219129658240384333408
716815792364272224441629276804197128965972936

All 15 rows

Technical details
Node id
n-ec890c07696bc147 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-f6fe0b69c7140161

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:   2;
  2:   8,     8;
  3:  20,   104,      24;
  4:  40,   584,     960,       64;
  5:  70,  2184,   11472,     7744,       160;
  6: 112,  6384,   76992,   191296,     58240,        384;
  7: 168, 15792,  364272,  2244416,   2927680,     419712,       896;
  8: 240, 34608, 1359168, 16815296,  59038080,   42578304,   2938880,     2048;
  9: 330, 69168, 4266240, 92928704, 689611680, 1459492224, 599035136, 20154368, 4608;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(n+1+(2*n^3-4*n^2-2*n+4)*x+(n^5-3*n^4+n^3+3*n^2-2*n)*x^2)*P(n-2,x) + ((2*n^2-3*n-5)*x^2+(2*n^4-4*n^3-2*n^2+4*n)*x^3)*d/dx P(n-2,x) - ((n+1)*x^3+(n^3-n)*x^4)*d^2/dx^2 P(n-2,x) + (2*n+(2*n^3-4*n^2+2*n)*x)*P(n-1,x) - (2*n^2-5*n+3)*x^2*d/dx P(n-1,x) + (n-1)*x^3*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(n + 1 + (2*n^3 - 4*n^2 - 2*n + 4)*x + (n^5 - 3*n^4 + n^3 + 3*n^2 - 2*n)*x^2)*P[n - 2, x] + ((2*n^2 - 3*n - 5)*x^2 + (2*n^4 - 4*n^3 - 2*n^2 + 4*n)*x^3)*D[P[n - 2, x], x] - ((n + 1)*x^3 + (n^3 - n)*x^4)*D[P[n - 2, x], {x, 2}] + (2*n + (2*n^3 - 4*n^2 + 2*n)*x)*P[n - 1, x] - (2*n^2 - 5*n + 3)*x^2*D[P[n - 1, x], x] + (n - 1)*x^3*D[P[n - 1, x], {x, 2}])/(n - 1)]];
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] = 2; for(n = 2, N, my(i = n + 1); v[i] = (-(n + 1 + (2*n^3 - 4*n^2 - 2*n + 4)*x + (n^5 - 3*n^4 + n^3 + 3*n^2 - 2*n)*x^2)*v[i - 2] + ((2*n^2 - 3*n - 5)*x^2 + (2*n^4 - 4*n^3 - 2*n^2 + 4*n)*x^3)*deriv(v[i - 2], x) - ((n + 1)*x^3 + (n^3 - n)*x^4)*deriv(deriv(v[i - 2], x), x) + (2*n + (2*n^3 - 4*n^2 + 2*n)*x)*v[i - 1] - (2*n^2 - 5*n + 3)*x^2*deriv(v[i - 1], x) + (n - 1)*x^3*deriv(deriv(v[i - 1], x), x))/(n - 1)); 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.