Real-rooted atlas

A181543 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(n^{2} + n^{5} x\right)\,P_{n}(x) = \left(12\,x - 30\,n x + n^{2} + 20\,n^{2} x + 10\,n^{2} x^{2} - 18\,n^{3} x^{2} + n^{5} x + 10\,n^{5} x^{2}\right)\,P_{n-1}(x) + \left(6\,x + 24\,x^{2} + 3\,n x - 15\,n x^{2} + 30\,n^{2} x^{3} + 6\,n^{3} x^{2} - 6\,n^{3} x^{3} + 3\,n^{4} x^{2} - 15\,n^{4} x^{3}\right)\,P_{n-1}'(x) + \left(6\,x^{2} + 6\,x^{3} + 10\,n^{2} x^{3} + 10\,n^{2} x^{4} + 6\,n^{3} x^{3} + 6\,n^{3} x^{4}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1123
2116320
3154814140
4112864825651038
512503000400062567882
614321012532000168751296760736
71686277831715002143755556624018472320

All 15 rows

Technical details
Node id
n-85696c1481362941 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-c06bc352c6c59952

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,    2;
  2: 1,   16,      3;
  3: 1,   54,     81,       4;
  4: 1,  128,    648,     256,        5;
  5: 1,  250,   3000,    4000,      625,        6;
  6: 1,  432,  10125,   32000,    16875,     1296,       7;
  7: 1,  686,  27783,  171500,   214375,    55566,    2401,      8;
  8: 1, 1024,  65856,  702464,  1715000,  1053696,  153664,   4096,    9;
  9: 1, 1458, 139968, 2370816, 10001880, 12002256, 4148928, 373248, 6561, 10;
%F Formula
The row polynomials satisfy (n^2+n^5*x)*P(n,x) = (n^2+(n^5+20*n^2-30*n+12)*x+(10*n^5-18*n^3+10*n^2)*x^2)*P(n-1,x) + ((3*n+6)*x+(3*n^4+6*n^3-15*n+24)*x^2-(15*n^4+6*n^3-30*n^2)*x^3)*d/dx P(n-1,x) + (6*x^2+(6*n^3+10*n^2+6)*x^3+(6*n^3+10*n^2)*x^4)*d^2/dx^2 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[((n^2 + (n^5 + 20*n^2 - 30*n + 12)*x + (10*n^5 - 18*n^3 + 10*n^2)*x^2)*P[n - 1, x] + ((3*n + 6)*x + (3*n^4 + 6*n^3 - 15*n + 24)*x^2 - (15*n^4 + 6*n^3 - 30*n^2)*x^3)*D[P[n - 1, x], x] + (6*x^2 + (6*n^3 + 10*n^2 + 6)*x^3 + (6*n^3 + 10*n^2)*x^4)*D[P[n - 1, x], {x, 2}])/(n^2 + n^5*x)]];
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] = ((n^2 + (n^5 + 20*n^2 - 30*n + 12)*x + (10*n^5 - 18*n^3 + 10*n^2)*x^2)*v[i - 1] + ((3*n + 6)*x + (3*n^4 + 6*n^3 - 15*n + 24)*x^2 - (15*n^4 + 6*n^3 - 30*n^2)*x^3)*deriv(v[i - 1], x) + (6*x^2 + (6*n^3 + 10*n^2 + 6)*x^3 + (6*n^3 + 10*n^2)*x^4)*deriv(deriv(v[i - 1], x), x))/(n^2 + n^5*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. [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