Real-rooted atlas

A075497 → D+1 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(48 - 28\,x + 4\,x^{2} + 3\,x^{3}\right)\,P_{n-3}(x) + \left(-16\,x - 4\,x^{2} + 8\,x^{3}\right)\,P_{n-3}'(x) + \left(-8\,x^{2} + 4\,x^{3}\right)\,P_{n-3}''(x) + \left(-44 + 20\,x + x^{2}\right)\,P_{n-2}(x) + \left(8\,x + 8\,x^{2}\right)\,P_{n-2}'(x) + \left(4\,x^{2}\right)\,P_{n-2}''(x) + \left(12 - 3\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + x\)
\(P_{2}(x) = 18 + 10\,x + x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1415
21810129
38882181189
44566402322811357
524644936264052040110601
61372838080283368120101054189329
778208295072295008115696207201778701806553

All 15 rows

Technical details
Node id
n-38cb1bd2c71c1a9f · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-35cdc7685b139f0b

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): 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:       1;
  1:       4,        1;
  2:      18,       10,        1;
  3:      88,       82,       18,        1;
  4:     456,      640,      232,       28,       1;
  5:    2464,     4936,     2640,      520,      40,       1;
  6:   13728,    38080,    28336,     8120,    1010,      54,     1;
  7:   78208,   295072,   295008,   115696,   20720,    1778,    70,    1;
  8:  452736,  2298880,  3020032,  1561728,  378336,   46368,  2912,   88,   1;
  9: 2650624, 18004096, 30620160, 20373760, 6424320, 1055712, 94080, 4512, 108, 1;
%F Formula
The row polynomials satisfy P(n,x) = (48-28*x+4*x^2+3*x^3)*P(n-3,x) - (16*x+4*x^2-8*x^3)*d/dx P(n-3,x) - (8*x^2-4*x^3)*d^2/dx^2 P(n-3,x) - (44-20*x-x^2)*P(n-2,x) + (8*x+8*x^2)*d/dx P(n-2,x) + 4*x^2*d^2/dx^2 P(n-2,x) + (12-3*x)*P(n-1,x) for n >= 3, with P(0,x) = 1, P(1,x) = 4 + x, P(2,x) = 18 + 10*x + x^2.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + x;
P[2, x_] := 18 + 10*x + x^2;
P[n_, x_] := P[n, x] = Expand[(48 - 28*x + 4*x^2 + 3*x^3)*P[n - 3, x] - (16*x + 4*x^2 - 8*x^3)*D[P[n - 3, x], x] - (8*x^2 - 4*x^3)*D[P[n - 3, x], {x, 2}] - (44 - 20*x - x^2)*P[n - 2, x] + (8*x + 8*x^2)*D[P[n - 2, x], x] + 4*x^2*D[P[n - 2, x], {x, 2}] + (12 - 3*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 4 + x; v[3] = 18 + 10*x + x^2; for(n = 3, N, my(i = n + 1); v[i] = (48 - 28*x + 4*x^2 + 3*x^3)*v[i - 3] - (16*x + 4*x^2 - 8*x^3)*deriv(v[i - 3], x) - (8*x^2 - 4*x^3)*deriv(deriv(v[i - 3], x), x) - (44 - 20*x - x^2)*v[i - 2] + (8*x + 8*x^2)*deriv(v[i - 2], x) + 4*x^2*deriv(deriv(v[i - 2], x), x) + (12 - 3*x)*v[i - 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. [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.