Real-rooted atlas

A075500 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-50 + 15\,x + 4\,x^{2}\right)\,P_{n-2}(x) + \left(-25\,x + 20\,x^{2}\right)\,P_{n-2}'(x) + \left(15 - 3\,x\right)\,P_{n-1}(x) + \left(5\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 40 + 17\,x + x^{2}\)
\(P_{1}(x) = 300 + 235\,x + 33\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
04017158
1300235331569
2250031257755416455
3225004187516125192580182506
421250057312531937557750402511111166887
520625008021875620812515881251645007490147118052763
620312500114453125120109375416587505998125400050128101881302944924
7201562500165742187523275781251064453125204225000186768758662502055023415474804535

All 15 rows

Technical details
Node id
n-6c1a403f2d2498fd · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-56f1d64aef9e5162

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:          40,           17,            1;
  1:         300,          235,           33,            1;
  2:        2500,         3125,          775,           54,            1;
  3:       22500,        41875,        16125,         1925,           80,           1;
  4:      212500,       573125,       319375,        57750,         4025,         111,          1;
  5:     2062500,      8021875,      6208125,      1588125,       164500,        7490,        147,         1;
  6:    20312500,    114453125,    120109375,     41658750,      5998125,      400050,      12810,       188,       1;
  7:   201562500,   1657421875,   2327578125,   1064453125,    204225000,    18676875,     866250,     20550,     234,     1;
  8:  2007812500,  24264453125,  45293359375,  26807343750,   6659640625,   800848125,   50386875,   1716000,   31350,   285,   1;
  9: 20039062500, 357982421875, 885598828125, 669930078125, 211175937500, 32445737500, 2656355625, 121666875, 3168000, 45925, 341, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(50-15*x-4*x^2)*P(n-2,x) - (25*x-20*x^2)*d/dx P(n-2,x) + (15-3*x)*P(n-1,x) + 5*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 40 + 17*x + x^2, P(1,x) = 300 + 235*x + 33*x^2 + x^3.
%t Mathematica
P[0, x_] := 40 + 17*x + x^2;
P[1, x_] := 300 + 235*x + 33*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[-(50 - 15*x - 4*x^2)*P[n - 2, x] - (25*x - 20*x^2)*D[P[n - 2, x], x] + (15 - 3*x)*P[n - 1, x] + 5*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 40 + 17*x + x^2; v[2] = 300 + 235*x + 33*x^2 + x^3; for(n = 2, N, my(i = n + 1); v[i] = -(50 - 15*x - 4*x^2)*v[i - 2] - (25*x - 20*x^2)*deriv(v[i - 2], x) + (15 - 3*x)*v[i - 1] + 5*x*deriv(v[i - 1], 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

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.