Real-rooted atlas

A166979 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(8 + 16\,x - x^{2}\right)\,P_{n-2}(x) + \left(72\,x - 9\,x^{2}\right)\,P_{n-2}'(x) + \left(-7 + 2\,x\right)\,P_{n-1}(x) + \left(9\,x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1 + 22\,x + 3\,x^{2}\)
\(P_{2}(x) = 1 + 222\,x + 90\,x^{2} + 4\,x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
1122326
21222904317
312222204323254503
4122222421509220475674074
512222228341833143602910084671400719
6122222221618281099143241469650738361372829864223
712222222231080672329917815266769955516003616160220809704300780
812222222225938661070879139722028444610253174856021490815231780829971018129456107

All 15 rows

Technical details
Node id
n-8a263f617dd30ab3 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-98d1f19ba42a5ad7

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 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 1,          22,             3;
   2: 1,         222,            90,             4;
   3: 1,        2222,          2043,           232,             5;
   4: 1,       22222,         42150,          9220,           475,            6;
   5: 1,      222222,        834183,        314360,         29100,          846,           7;
   6: 1,     2222222,      16182810,       9914324,       1469650,        73836,        1372,          8;
   7: 1,    22222222,     310806723,     299178152,      66769955,      5160036,      161602,       2080,        9;
   8: 1,   222222222,    5938661070,    8791397220,    2844461025,    317485602,    14908152,     317808,     2997,     10;
   9: 1,  2222222222,  113167893663,  254077336920,  116234304450,  18017690922,  1190348229,   37377600,   576315,   4150,   11;
  10: 1, 22222222222, 2153523312930, 7265055958644, 4618265935800, 968294947752, 86489792004, 3752564376, 84120795, 980650, 5566, 12;
%F Formula
The row polynomials satisfy P(n,x) = (8+16*x-x^2)*P(n-2,x) + (72*x-9*x^2)*d/dx P(n-2,x) - (7-2*x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1 + 22*x + 3*x^2, P(2,x) = 1 + 222*x + 90*x^2 + 4*x^3.
%t Mathematica
P[1, x_] := 1 + 22*x + 3*x^2;
P[2, x_] := 1 + 222*x + 90*x^2 + 4*x^3;
P[n_, x_] := P[n, x] = Expand[(8 + 16*x - x^2)*P[n - 2, x] + (72*x - 9*x^2)*D[P[n - 2, x], x] - (7 - 2*x)*P[n - 1, x] + 9*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1 + 22*x + 3*x^2; v[2] = 1 + 222*x + 90*x^2 + 4*x^3; for(n = 3, N, my(i = n); v[i] = (8 + 16*x - x^2)*v[i - 2] + (72*x - 9*x^2)*deriv(v[i - 2], x) - (7 - 2*x)*v[i - 1] + 9*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, v[i])))};
T(10)

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.