Real-rooted atlas

A391403 → x ↦ x+1 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(6\,n + 5\,n x + 4\,n^{2} x + n^{2} x^{2} + n^{3} x^{2}\right)\,P_{n}(x) = \left(-24 - 56\,x - 40\,x^{2} - 8\,x^{3} + 30\,n + 39\,n x + 4\,n x^{2} - 4\,n x^{3} + 26\,n^{2} x + 33\,n^{2} x^{2} + 8\,n^{2} x^{3} + 5\,n^{3} x^{2} + 4\,n^{3} x^{3}\right)\,P_{n-1}(x) + \left(-18 - 72\,x - 106\,x^{2} - 68\,x^{3} - 16\,x^{4} - 24\,n x - 74\,n x^{2} - 74\,n x^{3} - 24\,n x^{4} - 6\,n^{2} x^{2} - 14\,n^{2} x^{3} - 8\,n^{2} x^{4}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2347
371219
419481885
55116090301
6141540450801211
7393176418905604607
8110757127560336035018089
93139182882872816800315070105
1089535814010584077280220501512273775
1125653183700380160332640127050166321065835
12737895776321339470136752065835013305664684156285
1321294118089764646070542256031531508648648408416192645
14616227564600415909894209008801426425049729687567562745663094435
Technical details
Node id
n-6fc79809541bd4b3 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-975d9965822ff143

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7 (half 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;
  2:    3,     4;
  3:    7,    12;
  4:   19,    48,    18;
  5:   51,   160,    90;
  6:  141,   540,   450,    80;
  7:  393,  1764,  1890,   560;
  8: 1107,  5712,  7560,  3360,  350;
  9: 3139, 18288, 28728, 16800, 3150;
%F Formula
The row polynomials satisfy (6*n+(4*n^2+5*n)*x+(n^3+n^2)*x^2)*P(n,x) = (30*n-24+(26*n^2+39*n-56)*x+(5*n^3+33*n^2+4*n-40)*x^2+(4*n^3+8*n^2-4*n-8)*x^3)*P(n-1,x) - (18+(24*n+72)*x+(6*n^2+74*n+106)*x^2+(14*n^2+74*n+68)*x^3+(8*n^2+24*n+16)*x^4)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((30*n - 24 + (26*n^2 + 39*n - 56)*x + (5*n^3 + 33*n^2 + 4*n - 40)*x^2 + (4*n^3 + 8*n^2 - 4*n - 8)*x^3)*P[n - 1, x] - (18 + (24*n + 72)*x + (6*n^2 + 74*n + 106)*x^2 + (14*n^2 + 74*n + 68)*x^3 + (8*n^2 + 24*n + 16)*x^4)*D[P[n - 1, x], x])/(6*n + (4*n^2 + 5*n)*x + (n^3 + n^2)*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; for(n = 1, N, my(i = n + 1); v[i] = ((30*n - 24 + (26*n^2 + 39*n - 56)*x + (5*n^3 + 33*n^2 + 4*n - 40)*x^2 + (4*n^3 + 8*n^2 - 4*n - 8)*x^3)*v[i - 1] - (18 + (24*n + 72)*x + (6*n^2 + 74*n + 106)*x^2 + (14*n^2 + 74*n + 68)*x^3 + (8*n^2 + 24*n + 16)*x^4)*deriv(v[i - 1], x))/(6*n + (4*n^2 + 5*n)*x + (n^3 + n^2)*x^2)); 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

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  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.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023