Real-rooted atlas

A257626 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(1 + n\right)\,P_{n}(x) = \left(-36 - 18\,n - 36\,n x - 18\,n x^{2} - 18\,n^{2} x - 27\,n^{2} x^{2} - 9\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(-18\,x + 18\,x^{3} - 9\,n x - 18\,n x^{2} + 27\,n x^{3} - 9\,n^{2} x^{2} + 9\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(12 + 6\,x + 9\,n + 12\,n x + 6\,n^{2} x\right)\,P_{n-1}(x) + \left(3\,x - 3\,x^{2} + 3\,n x - 3\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 12\,x\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
161218
236216108360
3216280842128648100
4129631752101088635046480204120
5777633242419274762569968831060466565715360
64665633184083210516077099040535086009955224326592176359680
7279936321051604897917721927009440240876180097958354411236806022394885952139200

All 15 rows

Technical details
Node id
n-5ed42a17b514c663 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-7c3b635378939853

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:        6,         12;
  2:       36,        216,         108;
  3:      216,       2808,        4212,          864;
  4:     1296,      31752,      101088,        63504,          6480;
  5:     7776,     332424,     1927476,      2569968,        831060,         46656;
  6:    46656,    3318408,    32105160,     77099040,      53508600,       9955224,        326592;
  7:   279936,   32105160,   489791772,   1927009440,    2408761800,     979583544,     112368060,      2239488;
  8:  1679616,  304062984,  7033287024,  42619311216,   86715424800,   63928966824,   16411003056,   1216251936,    15116544;
  9: 10077696, 2837343816, 96713995140, 864354853008, 2679634565820, 3215561478984, 1512620992764, 257903987040, 12768047172, 100776960;
%F Formula
The row polynomials satisfy (n+1)*P(n,x) = -(18*n+36+(18*n^2+36*n)*x+(9*n^3+27*n^2+18*n)*x^2)*P(n-2,x) - ((9*n+18)*x+(9*n^2+18*n)*x^2-(9*n^2+27*n+18)*x^3)*d/dx P(n-2,x) + (9*n+12+(6*n^2+12*n+6)*x)*P(n-1,x) + ((3*n+3)*x-(3*n+3)*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 12*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 12*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(18*n + 36 + (18*n^2 + 36*n)*x + (9*n^3 + 27*n^2 + 18*n)*x^2)*P[n - 2, x] - ((9*n + 18)*x + (9*n^2 + 18*n)*x^2 - (9*n^2 + 27*n + 18)*x^3)*D[P[n - 2, x], x] + (9*n + 12 + (6*n^2 + 12*n + 6)*x)*P[n - 1, x] + ((3*n + 3)*x - (3*n + 3)*x^2)*D[P[n - 1, x], x])/(n + 1)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 12*x; for(n = 2, N, my(i = n + 1); v[i] = (-(18*n + 36 + (18*n^2 + 36*n)*x + (9*n^3 + 27*n^2 + 18*n)*x^2)*v[i - 2] - ((9*n + 18)*x + (9*n^2 + 18*n)*x^2 - (9*n^2 + 27*n + 18)*x^3)*deriv(v[i - 2], x) + (9*n + 12 + (6*n^2 + 12*n + 6)*x)*v[i - 1] + ((3*n + 3)*x - (3*n + 3)*x^2)*deriv(v[i - 1], x))/(n + 1)); 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. [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
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.