Real-rooted atlas

A286785 → x ↦ x+1

nonnegative coefficients

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

Recurrence

\(\left(56 + 30\,n + 4\,n^{2}\right)\,P_{n}(x) = \left(240 + 84\,x + 162\,n + 66\,n x + 27\,n^{2} + 12\,n^{2} x\right)\,P_{n-1}(x) + \left(-30\,x - 28\,x^{2} - 9\,n x - 8\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 30 + 18\,x + 2\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
03018250
1143132332310
27289103645222056
338766120340080075214273
4213184069829070969015301022102410
51201752691922355431024102327526601332753390
66906901776060184184099176029752849588431216825651948
74032015117058501404702090417603390660753480966006624207243074218

All 15 rows

Technical details
Node id
n-664dba754f71e70d · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-364ead120df7d525

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:        30,        18,         2;
  1:       143,       132,        33,         2;
  2:       728,       910,       364,        52,         2;
  3:      3876,      6120,      3400,       800,        75,         2;
  4:     21318,     40698,     29070,      9690,      1530,       102,        2;
  5:    120175,    269192,    235543,    102410,     23275,      2660,      133,       2;
  6:    690690,   1776060,   1841840,    991760,    297528,     49588,     4312,     168,      2;
  7:   4032015,  11705850,  14047020,   9041760,   3390660,    753480,    96600,    6624,    207,     2;
  8:  23841480,  77134200, 105183000,  78887250,  35626500,   9975420,  1719900,  175500,   9750,   250,   2;
  9: 142498692, 508373712, 776682060, 665727480, 352443960, 119617344, 26166294, 3617460, 301455, 13860, 297, 2;
%F Formula
The row polynomials satisfy (4*n^2+30*n+56)*P(n,x) = (27*n^2+162*n+240+(12*n^2+66*n+84)*x)*P(n-1,x) - ((9*n+30)*x+(8*n+28)*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 30 + 18*x + 2*x^2.
%t Mathematica
P[0, x_] := 30 + 18*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((27*n^2 + 162*n + 240 + (12*n^2 + 66*n + 84)*x)*P[n - 1, x] - ((9*n + 30)*x + (8*n + 28)*x^2)*D[P[n - 1, x], x])/(4*n^2 + 30*n + 56)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 30 + 18*x + 2*x^2; for(n = 1, N, my(i = n + 1); v[i] = ((27*n^2 + 162*n + 240 + (12*n^2 + 66*n + 84)*x)*v[i - 1] - ((9*n + 30)*x + (8*n + 28)*x^2)*deriv(v[i - 1], x))/(4*n^2 + 30*n + 56)); 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

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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.