Real-rooted atlas

A008292 → γ (shift +1) → x ↦ x+1 → D

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(18 + 48\,x + 32\,x^{2} - 38\,n - 92\,n x - 56\,n x^{2} + 24\,n^{2} + 52\,n^{2} x + 28\,n^{2} x^{2} - 4\,n^{3} - 8\,n^{3} x - 4\,n^{3} x^{2}\right)\,P_{n-2}(x) + \left(9 + 33\,x + 40\,x^{2} + 16\,x^{3} - 15\,n - 53\,n x - 62\,n x^{2} - 24\,n x^{3} + 6\,n^{2} + 20\,n^{2} x + 22\,n^{2} x^{2} + 8\,n^{2} x^{3}\right)\,P_{n-2}'(x) + \left(19 + 24\,x - 17\,n - 20\,n x + 4\,n^{2} + 4\,n^{2} x\right)\,P_{n-1}(x) + \left(6 + 14\,x + 8\,x^{2} - 3\,n - 7\,n x - 4\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 0\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
00
10
222
388
4543286
5324272596
6237030728166258
718288299521190460144
815838232428819814431744712558
91480572358516828131847075848586508
10150901384250275241782896161372161768960117281962
11165471336529706304619814592311496192559206401682409064
121950151302700543728095599958406035665920166517376013420953626350633638
132454676261297705887120152049484800114870850560416917785605711271936436576035588
143290069626741437937178688251651820600022230848563201016682324480214511726592133263011847751067555938
Technical details
Node id
n-26414d612486a2d9 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-32705c359d67320a

Observed on the stored rows ?

  • Degrees: -1, -1, 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6 (half 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:       0;
  1:       0;
  2:       2;
  3:       8;
  4:      54,      32;
  5:     324,     272;
  6:    2370,    3072,     816;
  7:   18288,   29952,   11904;
  8:  158382,  324288,  198144,  31744;
  9: 1480572, 3585168, 2813184, 707584;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = -(4*n^3-24*n^2+38*n-18+(8*n^3-52*n^2+92*n-48)*x+(4*n^3-28*n^2+56*n-32)*x^2)*P(n-2,x) + (6*n^2-15*n+9+(20*n^2-53*n+33)*x+(22*n^2-62*n+40)*x^2+(8*n^2-24*n+16)*x^3)*d/dx P(n-2,x) + (4*n^2-17*n+19+(4*n^2-20*n+24)*x)*P(n-1,x) - (3*n-6+(7*n-14)*x+(4*n-8)*x^2)*d/dx P(n-1,x) for n >= 3, with P(0,x) = 0, P(1,x) = 0, P(2,x) = 2.
%t Mathematica
P[0, x_] := 0;
P[1, x_] := 0;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 24*n^2 + 38*n - 18 + (8*n^3 - 52*n^2 + 92*n - 48)*x + (4*n^3 - 28*n^2 + 56*n - 32)*x^2)*P[n - 2, x] + (6*n^2 - 15*n + 9 + (20*n^2 - 53*n + 33)*x + (22*n^2 - 62*n + 40)*x^2 + (8*n^2 - 24*n + 16)*x^3)*D[P[n - 2, x], x] + (4*n^2 - 17*n + 19 + (4*n^2 - 20*n + 24)*x)*P[n - 1, x] - (3*n - 6 + (7*n - 14)*x + (4*n - 8)*x^2)*D[P[n - 1, x], x])/(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] = 0; v[3] = 2; for(n = 3, N, my(i = n + 1); v[i] = (-(4*n^3 - 24*n^2 + 38*n - 18 + (8*n^3 - 52*n^2 + 92*n - 48)*x + (4*n^3 - 28*n^2 + 56*n - 32)*x^2)*v[i - 2] + (6*n^2 - 15*n + 9 + (20*n^2 - 53*n + 33)*x + (22*n^2 - 62*n + 40)*x^2 + (8*n^2 - 24*n + 16)*x^3)*deriv(v[i - 2], x) + (4*n^2 - 17*n + 19 + (4*n^2 - 20*n + 24)*x)*v[i - 1] - (3*n - 6 + (7*n - 14)*x + (4*n - 8)*x^2)*deriv(v[i - 1], x))/(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. [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.