Real-rooted atlas

A173018 → reverse (shift +1) → θ+1 → x ↦ x+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernel

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1224
258316
3183824484
484220195645568
54801500171084416064700
632401176016380108083325384745904
72520010416017136014162460690123548968516292
8221760102816019504801937376106722031602044135204896567208
92177280111888002404080027891360187992007351380156303015336846081093169836
102358720013305600031933440042378336033804540016475976047572140745201652222510240111458122752
11279417600171642240045505152006798496320626610600036633801601343606880294196848345785221750990225281224948493460
1235925120002387024640069275606400115061506560120345825600821348985603663563904010390091712175859283615718430058007954915213463227953368
13498161664003559780224001122420499200205159578624024010354368001874679206400986406220800345035570880771674904001024151310070321251019033572106496149275098265212
14741015475200566658892800019288196928000384746858496004981876099200043809858892800265889752128001108878458688031029925726805554159884005843795958031066308246196560522937615199196882211760
Technical details
Node id
n-e547c10831e25b02 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-efe4fe8c256eb4ca

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:       2,        2;
  2:       5,        8,        3;
  3:      18,       38,       24,        4;
  4:      84,      220,      195,       64,        5;
  5:     480,     1500,     1710,      844,      160,       6;
  6:    3240,    11760,    16380,    10808,     3325,     384,       7;
  7:   25200,   104160,   171360,   141624,    60690,   12354,     896,      8;
  8:  221760,  1028160,  1950480,  1937376,  1067220,  316020,   44135,   2048,    9;
  9: 2177280, 11188800, 24040800, 27891360, 18799200, 7351380, 1563030, 153368, 4608, 10;
%F Formula
The row polynomials satisfy n*P(n,x) = -(n^3-2*n^2-n+2+(2*n^3-2*n^2-2*n+2)*x+(n^3-n)*x^2)*P(n-2,x) + ((n^2-1)*x+(2*n^2+n-1)*x^2+(n^2+n)*x^3)*d/dx P(n-2,x) + (2*n^2-n-1+2*n^2*x)*P(n-1,x) - (n*x+n*x^2)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 2 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(n^3 - 2*n^2 - n + 2 + (2*n^3 - 2*n^2 - 2*n + 2)*x + (n^3 - n)*x^2)*P[n - 2, x] + ((n^2 - 1)*x + (2*n^2 + n - 1)*x^2 + (n^2 + n)*x^3)*D[P[n - 2, x], x] + (2*n^2 - n - 1 + 2*n^2*x)*P[n - 1, x] - (n*x + n*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; v[2] = 2 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(n^3 - 2*n^2 - n + 2 + (2*n^3 - 2*n^2 - 2*n + 2)*x + (n^3 - n)*x^2)*v[i - 2] + ((n^2 - 1)*x + (2*n^2 + n - 1)*x^2 + (n^2 + n)*x^3)*deriv(v[i - 2], x) + (2*n^2 - n - 1 + 2*n^2*x)*v[i - 1] - (n*x + n*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. [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.