Real-rooted atlas

A228955 → D+1

nonnegative coefficients

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

Recurrence

\(\left(2 - 10\,x + 4\,x^{2} - n + 4\,n x + n^{2}\right)\,P_{n}(x) = \left(2\,n - 14\,n x - 2\,n x^{2} + 4\,n x^{3} + n^{2} + 3\,n^{2} x + 4\,n^{2} x^{2} + n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(-4\,x + 12\,x^{3} - 8\,x^{4} + 2\,n x + 6\,n x^{2} - 8\,n x^{3} + 2\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1213
28614
34248696
4264480120864
51920540021601209600
61584065520403205040126720
714616084672077616016128050401935360
814918401161216015240960483840036288033546240
91669248016873920030481920014152320018144000362880650280960
102032128002594592000622702080040715136007983360003991680013934592000
1126744256004215214080013040818560011591838720032931360000287400960039916800326998425600
123784112640072233441280028083863808003287866982400130144734720017435658240062270208008338459852800
135728859136001303315453440062332478208000934987173120004986598256640096332011776006102480384006227020800229552894771200
1492408988672002471504555520001427980409856000268029656294400018699743462400004997931434496004881984307200013076743680006784563334348800
Technical details
Node id
n-431f8937f41987e6 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-2cceb230b30e91a5

Observed on the stored rows ?

  • Degrees: 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 7, 7 (half 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:        1;
  1:        2,         1;
  2:        8,         6;
  3:       42,        48,         6;
  4:      264,       480,       120;
  5:     1920,      5400,      2160,       120;
  6:    15840,     65520,     40320,      5040;
  7:   146160,    846720,    776160,    161280,     5040;
  8:  1491840,  11612160,  15240960,   4838400,   362880;
  9: 16692480, 168739200, 304819200, 141523200, 18144000, 362880;
%F Formula
The row polynomials satisfy (n^2-n+2+(4*n-10)*x+4*x^2)*P(n,x) = (n^3+n^2+2*n+(n^3+3*n^2-14*n)*x+(4*n^2-2*n)*x^2+4*n*x^3)*P(n-1,x) + ((2*n^2+2*n-4)*x-(2*n^2-6*n)*x^2-(8*n-12)*x^3-8*x^4)*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[Together[((n^3 + n^2 + 2*n + (n^3 + 3*n^2 - 14*n)*x + (4*n^2 - 2*n)*x^2 + 4*n*x^3)*P[n - 1, x] + ((2*n^2 + 2*n - 4)*x - (2*n^2 - 6*n)*x^2 - (8*n - 12)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(n^2 - n + 2 + (4*n - 10)*x + 4*x^2)]];
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] = ((n^3 + n^2 + 2*n + (n^3 + 3*n^2 - 14*n)*x + (4*n^2 - 2*n)*x^2 + 4*n*x^3)*v[i - 1] + ((2*n^2 + 2*n - 4)*x - (2*n^2 - 6*n)*x^2 - (8*n - 12)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(n^2 - n + 2 + (4*n - 10)*x + 4*x^2)); 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. [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
  2. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.