Real-rooted atlas

A243662 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-2\,n + 2\,n^{2} - 2\,n^{2} x + 12\,n^{3} - 2\,n^{3} x + 4\,n^{4} x\right)\,P_{n}(x) = \left(-6 - 8\,x - 4\,x^{2} + 45\,n + 50\,n x + 14\,n x^{2} - 108\,n^{2} - 87\,n^{2} x - 10\,n^{2} x^{2} + 81\,n^{3} + 18\,n^{3} x - 12\,n^{3} x^{2} + 27\,n^{4} x + 12\,n^{4} x^{2}\right)\,P_{n-1}(x) + \left(-3\,x - 4\,x^{2} - 2\,x^{3} + 18\,n x + 19\,n x^{2} + 6\,n x^{3} - 27\,n^{2} x - 24\,n^{2} x^{2} - 2\,n^{2} x^{3} - 9\,n^{3} x^{2} - 8\,n^{3} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2325
31216331
455110454214
52737284689651570
61428476042001400175611969
777523100834884163203400288793659
8432632018942765071675804987571824418746750

All 15 rows

Technical details
Node id
n-368af004dbd4e002 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-58a400439b00a083

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 -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;
   2:       3,       2;
   3:      12,      16,        3;
   4:      55,     110,       45,        4;
   5:     273,     728,      468,       96,       5;
   6:    1428,    4760,     4200,     1400,     175,       6;
   7:    7752,   31008,    34884,    16320,    3400,     288,      7;
   8:   43263,  201894,   276507,   167580,   49875,    7182,    441,     8;
   9:  246675, 1315600,  2125200,  1586816,  619850,  129360,  13720,   640,   9;
  10: 1430715, 8584290, 15984540, 14208480, 6906900, 1912680, 297528, 24288, 891, 10;
%F Formula
The row polynomials satisfy (12*n^3+2*n^2-2*n+(4*n^4-2*n^3-2*n^2)*x)*P(n,x) = (81*n^3-108*n^2+45*n-6+(27*n^4+18*n^3-87*n^2+50*n-8)*x+(12*n^4-12*n^3-10*n^2+14*n-4)*x^2)*P(n-1,x) - ((27*n^2-18*n+3)*x+(9*n^3+24*n^2-19*n+4)*x^2+(8*n^3+2*n^2-6*n+2)*x^3)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((81*n^3 - 108*n^2 + 45*n - 6 + (27*n^4 + 18*n^3 - 87*n^2 + 50*n - 8)*x + (12*n^4 - 12*n^3 - 10*n^2 + 14*n - 4)*x^2)*P[n - 1, x] - ((27*n^2 - 18*n + 3)*x + (9*n^3 + 24*n^2 - 19*n + 4)*x^2 + (8*n^3 + 2*n^2 - 6*n + 2)*x^3)*D[P[n - 1, x], x])/(12*n^3 + 2*n^2 - 2*n + (4*n^4 - 2*n^3 - 2*n^2)*x)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((81*n^3 - 108*n^2 + 45*n - 6 + (27*n^4 + 18*n^3 - 87*n^2 + 50*n - 8)*x + (12*n^4 - 12*n^3 - 10*n^2 + 14*n - 4)*x^2)*v[i - 1] - ((27*n^2 - 18*n + 3)*x + (9*n^3 + 24*n^2 - 19*n + 4)*x^2 + (8*n^3 + 2*n^2 - 6*n + 2)*x^3)*deriv(v[i - 1], x))/(12*n^3 + 2*n^2 - 2*n + (4*n^4 - 2*n^3 - 2*n^2)*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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.