Real-rooted atlas

A114656 → basis A013610 → basis A013610

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(8 + 72\,x + 162\,x^{2} - 4\,n - 36\,n x - 81\,n x^{2}\right)\,P_{n-2}(x) + \left(-6 - 9\,x + 12\,n + 18\,n x\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 6 + 9\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
26915
34412681251
4360151219447294545
53152174243369626244656187077
628896198000511920612360328050590491738275
727385622386247251120117514809841500389723453144135785255
826615042527257698413056201670560231472080145496736446410444782969754410525

All 15 rows

Technical details
Node id
n-246f6d36a4b25acd · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-34f66af6eaa7ad99

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:         6,          9;
   3:        44,        126,          81;
   4:       360,       1512,        1944,         729;
   5:      3152,      17424,       33696,       26244,        6561;
   6:     28896,     198000,      511920,      612360,      328050,       59049;
   7:    273856,    2238624,     7251120,    11751480,     9841500,     3897234,      531441;
   8:   2661504,   25272576,    98413056,   201670560,   231472080,   145496736,    44641044,     4782969;
   9:  26380544,  285331968,  1298239488,  3220197120,  4711742784,  4100362560,  2023727328,   497428776,   43046721;
  10: 265655808, 3223971072, 16785335808, 48902579712, 87139317888, 97376997312, 67140130176, 26861153904, 5423886846, 387420489;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = -(4*n-8+(36*n-72)*x+(81*n-162)*x^2)*P(n-2,x) + (12*n-6+(18*n-9)*x)*P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 6 + 9*x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 6 + 9*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n - 8 + (36*n - 72)*x + (81*n - 162)*x^2)*P[n - 2, x] + (12*n - 6 + (18*n - 9)*x)*P[n - 1, x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 6 + 9*x; for(n = 3, N, my(i = n); v[i] = (-(4*n - 8 + (36*n - 72)*x + (81*n - 162)*x^2)*v[i - 2] + (12*n - 6 + (18*n - 9)*x)*v[i - 1])/(n + 1)); 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. [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