Real-rooted atlas

A008459 → basis A013610 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1325
2918633
3271089020245
481540810420701921
5243243056705040189025215525
6729102063402045360283508316924127905
72187408241837083402003118501496883603634321067925

All 15 rows

Technical details
Node id
n-a0eb494bfea0eebf · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-95d48315f211033d

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:     3,      2;
  2:     9,     18,       6;
  3:    27,    108,      90,       20;
  4:    81,    540,     810,      420,       70;
  5:   243,   2430,    5670,     5040,     1890,      252;
  6:   729,  10206,   34020,    45360,    28350,     8316,      924;
  7:  2187,  40824,  183708,   340200,   311850,   149688,    36036,    3432;
  8:  6561, 157464,  918540,  2245320,  2806650,  1945944,   756756,  154440,  12870;
  9: 19683, 590490, 4330260, 13471920, 21891870, 20432412, 11351340, 3706560, 656370, 48620;
%F Formula
The row polynomials satisfy n*P(n,x) = (3*n+2*n*x)*P(n-1,x) + (6*x+2*x^2)*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[((3*n + 2*n*x)*P[n - 1, x] + (6*x + 2*x^2)*D[P[n - 1, x], x])/(n)]];
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] = ((3*n + 2*n*x)*v[i - 1] + (6*x + 2*x^2)*deriv(v[i - 1], x))/(n)); 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