Real-rooted atlas

A062264 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(24\,n - 16\,n x + 4\,n x^{2} - 2\,n^{2} + 4\,n^{2} x + n^{2} x^{2} + 2\,n^{3} + 2\,n^{3} x + n^{4}\right)\,P_{n}(x) = \left(80 - 80\,x + 28\,x^{2} - 4\,x^{3} + 40\,n + 24\,n x - 18\,n x^{2} + 6\,n x^{3} + 21\,n^{2} + 14\,n^{2} x + 4\,n^{2} x^{2} + 3\,n^{2} x^{3} + 8\,n^{3} + 4\,n^{3} x + 6\,n^{3} x^{2} + n^{4} + 3\,n^{4} x\right)\,P_{n-1}(x) + \left(16\,x - 24\,x^{2} + 12\,x^{3} - 4\,x^{4} + 8\,n x - 4\,n x^{2} - 2\,n x^{3} - 2\,n x^{4} + 3\,n^{2} x + n^{2} x^{2} - 4\,n^{2} x^{3} + 2\,n^{3} x - 2\,n^{3} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1617
22714142
398105241228
42945602643611155
5756231019205405015577
61722781210350510097566126026
73564226384435234650115501617841118456

All 15 rows

Technical details
Node id
n-8926f861ec62bbf2 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-b948c3794f60a9b4

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,      1;
  2:    27,     14,      1;
  3:    98,    105,     24,      1;
  4:   294,    560,    264,     36,      1;
  5:   756,   2310,   1920,    540,     50,      1;
  6:  1722,   7812,  10350,   5100,    975,     66,     1;
  7:  3564,  22638,  44352,  34650,  11550,   1617,    84,    1;
  8:  6831,  58080, 158928, 182952,  96250,  23408,  2520,  104,   1;
  9: 12298, 135135, 494208, 792792, 612612, 234234, 43680, 3744, 126, 1;
%F Formula
The row polynomials satisfy (n^4+2*n^3-2*n^2+24*n+(2*n^3+4*n^2-16*n)*x+(n^2+4*n)*x^2)*P(n,x) = (n^4+8*n^3+21*n^2+40*n+80+(3*n^4+4*n^3+14*n^2+24*n-80)*x+(6*n^3+4*n^2-18*n+28)*x^2+(3*n^2+6*n-4)*x^3)*P(n-1,x) + ((2*n^3+3*n^2+8*n+16)*x-(2*n^3-n^2+4*n+24)*x^2-(4*n^2+2*n-12)*x^3-(2*n+4)*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^4 + 8*n^3 + 21*n^2 + 40*n + 80 + (3*n^4 + 4*n^3 + 14*n^2 + 24*n - 80)*x + (6*n^3 + 4*n^2 - 18*n + 28)*x^2 + (3*n^2 + 6*n - 4)*x^3)*P[n - 1, x] + ((2*n^3 + 3*n^2 + 8*n + 16)*x - (2*n^3 - n^2 + 4*n + 24)*x^2 - (4*n^2 + 2*n - 12)*x^3 - (2*n + 4)*x^4)*D[P[n - 1, x], x])/(n^4 + 2*n^3 - 2*n^2 + 24*n + (2*n^3 + 4*n^2 - 16*n)*x + (n^2 + 4*n)*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^4 + 8*n^3 + 21*n^2 + 40*n + 80 + (3*n^4 + 4*n^3 + 14*n^2 + 24*n - 80)*x + (6*n^3 + 4*n^2 - 18*n + 28)*x^2 + (3*n^2 + 6*n - 4)*x^3)*v[i - 1] + ((2*n^3 + 3*n^2 + 8*n + 16)*x - (2*n^3 - n^2 + 4*n + 24)*x^2 - (4*n^2 + 2*n - 12)*x^3 - (2*n + 4)*x^4)*deriv(v[i - 1], x))/(n^4 + 2*n^3 - 2*n^2 + 24*n + (2*n^3 + 4*n^2 - 16*n)*x + (n^2 + 4*n)*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

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.