Real-rooted atlas

A075498 → reverse

nonnegative coefficients

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1134
219919
31186327109
413022540581742
5145585243025112435815
616312609450243811530972951193
7184239428350137781234738925832187498118
811084158714425630311888110220522555768565615296321
911356750158760184898710333575248625452040471033526711968360987817
101165103953207605182947436153411798661703187552501869950612013570959049754940848
11119815345601425128811871524572289649423083016747800401205806117030912581208733031771479983845261
12123421879106177529119662461605716416654409620385709344492632750614984778799015448694625725416965531441140329768789
13127330303178378260972912124783659015245081280107883135360416206043253788796913905613622150505139763668590435303323115943232087182244308
14131540950287469011983771830770239504893666921047376508608026817518858138280505692459124455758590807503229474650126222605054126119793709478296932725315072135
Technical details
Node id
n-ca90c87faebb0901 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-ce17531d5a832eec

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,   3;
  2: 1,   9,    9;
  3: 1,  18,   63,     27;
  4: 1,  30,  225,    405,      81;
  5: 1,  45,  585,   2430,    2511,      243;
  6: 1,  63, 1260,   9450,   24381,    15309,      729;
  7: 1,  84, 2394,  28350,  137781,   234738,    92583,     2187;
  8: 1, 108, 4158,  71442,  563031,  1888110,  2205225,   557685,    6561;
  9: 1, 135, 6750, 158760, 1848987, 10333575, 24862545, 20404710, 3352671, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (1+3*n*x)*P(n-1,x) - 3*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[(1 + 3*n*x)*P[n - 1, x] - 3*x^2*D[P[n - 1, x], 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] = (1 + 3*n*x)*v[i - 1] - 3*x^2*deriv(v[i - 1], 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

Same fingerprint ?

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. [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.