Real-rooted atlas

A357613 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + 8\,n - 10\,n^{2} + 4\,n^{3}\right)\,P_{n}(x) = \left(-6 - 6\,x + 33\,n + 35\,n x + 2\,n x^{2} - 54\,n^{2} - 55\,n^{2} x - 3\,n^{2} x^{2} + 27\,n^{3} + 24\,n^{3} x + n^{3} x^{2}\right)\,P_{n-1}(x) + \left(-2\,x - 2\,x^{2} + 9\,n x + 10\,n x^{2} + n x^{3} - 9\,n^{2} x - 10\,n^{2} x^{2} - n^{2} x^{3}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
122
2201232
316821060438
41320252015122805632
51001025740237609240126070010
674256240240300300180180514805544852000
754264021162963341520272272012012002702702402410218670

All 15 rows

Technical details
Node id
n-0da20f282347d5c3 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-69f4d9beddd80e10

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (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:        0;
  1:        2;
  2:       20,        12;
  3:      168,       210,        60;
  4:     1320,      2520,      1512,       280;
  5:    10010,     25740,     23760,      9240,      1260;
  6:    74256,    240240,    300300,    180180,     51480,      5544;
  7:   542640,   2116296,   3341520,   2722720,   1201200,    270270,    24024;
  8:  3922512,  17907120,  34186320,  35271600,  21162960,   7351344,  1361360,  102960;
  9: 28120950, 147094200, 329491008, 411863760, 313374600, 148140720, 42325920, 6651216, 437580;
%F Formula
The row polynomials satisfy (4*n^3-10*n^2+8*n-2)*P(n,x) = (27*n^3-54*n^2+33*n-6+(24*n^3-55*n^2+35*n-6)*x+(n^3-3*n^2+2*n)*x^2)*P(n-1,x) - ((9*n^2-9*n+2)*x+(10*n^2-10*n+2)*x^2+(n^2-n)*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((27*n^3 - 54*n^2 + 33*n - 6 + (24*n^3 - 55*n^2 + 35*n - 6)*x + (n^3 - 3*n^2 + 2*n)*x^2)*P[n - 1, x] - ((9*n^2 - 9*n + 2)*x + (10*n^2 - 10*n + 2)*x^2 + (n^2 - n)*x^3)*D[P[n - 1, x], x])/(4*n^3 - 10*n^2 + 8*n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((27*n^3 - 54*n^2 + 33*n - 6 + (24*n^3 - 55*n^2 + 35*n - 6)*x + (n^3 - 3*n^2 + 2*n)*x^2)*v[i - 1] - ((9*n^2 - 9*n + 2)*x + (10*n^2 - 10*n + 2)*x^2 + (n^2 - n)*x^3)*deriv(v[i - 1], x))/(4*n^3 - 10*n^2 + 8*n - 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

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.