Real-rooted atlas

A075502 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-98 - 14\,x - x^{2}\right)\,P_{n-2}(x) + \left(-49\,x - 7\,x^{2}\right)\,P_{n-2}'(x) + \left(21 + 2\,x\right)\,P_{n-1}(x) + \left(7\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 21 + 2\,x\)
\(P_{1}(x) = 343 + 84\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8Pn(1)
021223
1343843430
25145245021047809
3744316174095554205146151
4105884114454023601502744073562892574
51494142332471124122523031440600651701176761171803
6210003465711776450391771170667574044537890135828176481384983979
7294581331115367312380120372574352858870700274038135121010402572502520933495652780

All 15 rows

Technical details
Node id
n-7be5b76137b0bab9 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-fa1c0a5250627916

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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:           21,             2;
  1:          343,            84,              3;
  2:         5145,          2450,            210,             4;
  3:        74431,         61740,           9555,           420,            5;
  4:      1058841,       1445402,         360150,         27440,          735,           6;
  5:     14941423,      32471124,       12252303,       1440600,        65170,        1176,          7;
  6:    210003465,     711776450,      391771170,      66757404,      4537890,      135828,       1764,        8;
  7:   2945813311,   15367312380,    12037257435,    2858870700,    274038135,    12101040,     257250,     2520,      9;
  8:  41281739961,  328605186602,   360094176750,  116110151080,  15083190045,   921796722,   28523880,   452760,   3465,   10;
  9: 578226834703, 6983272398564, 10575544728903, 4543980856800, 778631670740, 63267867432, 2672766789, 61122600, 751905, 4620, 11;
%F Formula
The row polynomials satisfy P(n,x) = -(98+14*x+x^2)*P(n-2,x) - (49*x+7*x^2)*d/dx P(n-2,x) + (21+2*x)*P(n-1,x) + 7*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 21 + 2*x, P(1,x) = 343 + 84*x + 3*x^2.
%t Mathematica
P[0, x_] := 21 + 2*x;
P[1, x_] := 343 + 84*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(98 + 14*x + x^2)*P[n - 2, x] - (49*x + 7*x^2)*D[P[n - 2, x], x] + (21 + 2*x)*P[n - 1, x] + 7*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] = 21 + 2*x; v[2] = 343 + 84*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(98 + 14*x + x^2)*v[i - 2] - (49*x + 7*x^2)*deriv(v[i - 2], x) + (21 + 2*x)*v[i - 1] + 7*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.