Real-rooted atlas

A241171 → θ+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-6\,x^{2}\right)\,P_{n-2}(x) + \left(-2\,x^{2} - 9\,x^{3}\right)\,P_{n-2}'(x) + \left(-x^{3} - 2\,x^{4}\right)\,P_{n-2}''(x) + \left(1 + 12\,x\right)\,P_{n-1}(x) + \left(3\,x + 11\,x^{2}\right)\,P_{n-1}'(x) + \left(x^{2} + 2\,x^{3}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111213
2160270331
3125237801008014113
41102039690302400567000910111
5140923801606320160311850004490640082795813
6116380351351011531520011351340004086482400476756280010108024291
7165532319901401976214240348648300002370159792006674587920006538371840001595185055113

All 15 rows

Technical details
Node id
n-3c9286c555c35626 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-403c190250d8ebb3

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: 1,      12;
  2: 1,      60,        270;
  3: 1,     252,       3780,        10080;
  4: 1,    1020,      39690,       302400,         567000;
  5: 1,    4092,     380160,      6320160,       31185000,        44906400;
  6: 1,   16380,    3513510,    115315200,     1135134000,      4086482400,       4767562800;
  7: 1,   65532,   31990140,   1976214240,    34864830000,    237015979200,     667458792000,      653837184000;
  8: 1,  262140,  289385730,  32813726400,   982782801000,  11293869787200,   57868677266400,   133382785536000,   112541725296000;
  9: 1, 1048572, 2610369720, 535823356320, 26415342135000, 484415710178400, 4034597695128000, 16472774013696000, 32074391709360000, 23758808673600000;
%F Formula
The row polynomials satisfy P(n,x) = -6*x^2*P(n-2,x) - (2*x^2+9*x^3)*d/dx P(n-2,x) - (x^3+2*x^4)*d^2/dx^2 P(n-2,x) + (1+12*x)*P(n-1,x) + (3*x+11*x^2)*d/dx P(n-1,x) + (x^2+2*x^3)*d^2/dx^2 P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 12*x;
P[n_, x_] := P[n, x] = Expand[-6*x^2*P[n - 2, x] - (2*x^2 + 9*x^3)*D[P[n - 2, x], x] - (x^3 + 2*x^4)*D[P[n - 2, x], {x, 2}] + (1 + 12*x)*P[n - 1, x] + (3*x + 11*x^2)*D[P[n - 1, x], x] + (x^2 + 2*x^3)*D[P[n - 1, x], {x, 2}]];
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] = 1 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = -6*x^2*v[i - 2] - (2*x^2 + 9*x^3)*deriv(v[i - 2], x) - (x^3 + 2*x^4)*deriv(deriv(v[i - 2], x), x) + (1 + 12*x)*v[i - 1] + (3*x + 11*x^2)*deriv(v[i - 1], x) + (x^2 + 2*x^3)*deriv(deriv(v[i - 1], x), 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.