Real-rooted atlas

basis-map coefficients of A030528, shifted → basis A013609 → D+1

only rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1-3-2-5
2512421
3-7-30-36-8-81
49561209616297
5-11-90-280-400-240-32-1053
61313254011201200576643645
7-15-182-924-2520-3920-3360-1344-128-12393

All 15 rows

Technical details
Node id
n-1d41671801583ee9 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4be2d3b20c0d29e0

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): no
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:   1;
  1:  -3,   -2;
  2:   5,   12,     4;
  3:  -7,  -30,   -36,    -8;
  4:   9,   56,   120,    96,     16;
  5: -11,  -90,  -280,  -400,   -240,    -32;
  6:  13,  132,   540,  1120,   1200,    576,     64;
  7: -15, -182,  -924, -2520,  -3920,  -3360,  -1344,   -128;
  8:  17,  240,  1456,  4928,  10080,  12544,   8960,   3072,   256;
  9: -19, -306, -2160, -8736, -22176, -36288, -37632, -23040, -6912, -512;
%F Formula
The row polynomials satisfy (2*n-1+2*x)*P(n,x) = -(2*n+1+(4*n+4)*x+4*x^2)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(2*n + 1 + (4*n + 4)*x + 4*x^2)*P[n - 1, x])/(2*n - 1 + 2*x)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(2*n + 1 + (4*n + 4)*x + 4*x^2)*v[i - 1])/(2*n - 1 + 2*x)); 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

Same fingerprint ?

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.