Real-rooted atlas

A201730 → D+1

nonnegative coefficients

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

Recurrence

\(\left(48 + 88\,x - 16\,x^{2} - 20\,n - 60\,n x + 8\,n x^{2} - 6\,n^{2} + 8\,n^{2} x + 2\,n^{3}\right)\,P_{n}(x) = \left(180 + 264\,x - 4\,x^{2} - 8\,x^{3} - 96\,n - 170\,n x - 6\,n x^{2} + 4\,n x^{3} - 7\,n^{2} + 17\,n^{2} x + 4\,n^{2} x^{2} + 5\,n^{3} + n^{3} x\right)\,P_{n-1}(x) + \left(-144 - 180\,x + 16\,x^{2} + 44\,x^{3} - 8\,x^{4} + 6\,n + 28\,n x + 14\,n x^{2} - 8\,n x^{3} + 6\,n^{2} + 4\,n^{2} x - 2\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 2\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
222
3617
420626
56728196
622212010352
77284896611284
823681918364144664
9765372961810120116880
10246022707283888161860896
1178730984253688048301901219056
1225095635184615558426004154022785952
1379715912400446347311304961060527612813312
142524342431797625187506198406523426002610048768
157971612148803699765154281607936887220447378135822912
Technical details
Node id
n-d9d841c751c18e23 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-1bbb46c3c18991cb

Observed on the stored rows ?

  • Degrees: 0, 0, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1:     1;
   2:     2;
   3:     6,     1;
   4:    20,     6;
   5:    67,    28,    1;
   6:   222,   120,   10;
   7:   728,   489,   66,   1;
   8:  2368,  1918,  364,  14;
   9:  7653,  7296, 1810, 120,  1;
  10: 24602, 27072, 8388, 816, 18;
%F Formula
The row polynomials satisfy (2*n^3-6*n^2-20*n+48+(8*n^2-60*n+88)*x+(8*n-16)*x^2)*P(n,x) = (5*n^3-7*n^2-96*n+180+(n^3+17*n^2-170*n+264)*x+(4*n^2-6*n-4)*x^2+(4*n-8)*x^3)*P(n-1,x) + (6*n^2+6*n-144+(4*n^2+28*n-180)*x-(2*n^2-14*n-16)*x^2-(8*n-44)*x^3-8*x^4)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 2.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 2;
P[n_, x_] := P[n, x] = Expand[Together[((5*n^3 - 7*n^2 - 96*n + 180 + (n^3 + 17*n^2 - 170*n + 264)*x + (4*n^2 - 6*n - 4)*x^2 + (4*n - 8)*x^3)*P[n - 1, x] + (6*n^2 + 6*n - 144 + (4*n^2 + 28*n - 180)*x - (2*n^2 - 14*n - 16)*x^2 - (8*n - 44)*x^3 - 8*x^4)*D[P[n - 1, x], x])/(2*n^3 - 6*n^2 - 20*n + 48 + (8*n^2 - 60*n + 88)*x + (8*n - 16)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 2; for(n = 3, N, my(i = n); v[i] = ((5*n^3 - 7*n^2 - 96*n + 180 + (n^3 + 17*n^2 - 170*n + 264)*x + (4*n^2 - 6*n - 4)*x^2 + (4*n - 8)*x^3)*v[i - 1] + (6*n^2 + 6*n - 144 + (4*n^2 + 28*n - 180)*x - (2*n^2 - 14*n - 16)*x^2 - (8*n - 44)*x^3 - 8*x^4)*deriv(v[i - 1], x))/(2*n^3 - 6*n^2 - 20*n + 48 + (8*n^2 - 60*n + 88)*x + (8*n - 16)*x^2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

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.