Real-rooted atlas

A128100 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-9\,n - 13\,n x + 12\,n x^{2} + 12\,n^{2} x + 3\,n^{3}\right)\,P_{n}(x) = \left(24 + 40\,x - 42\,x^{2} - 72\,x^{3} - 36\,n - 101\,n x - 18\,n x^{2} + 72\,n x^{3} - 9\,n^{2} + 42\,n^{2} x + 72\,n^{2} x^{2} + 15\,n^{3} + 18\,n^{3} x\right)\,P_{n-1}(x) + \left(-12 - 100\,x - 283\,x^{2} - 336\,x^{3} - 144\,x^{4} - 36\,n - 174\,n x - 276\,n x^{2} - 144\,n x^{3} - 18\,n^{2} - 51\,n^{2} x - 36\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
111
2639
311617
43239980
56910227198
616633918927721
73678946301081999
882824392349810816507
9181761747209324040518845
103986155072189713365324024358238
118643377346188446332149851458171036
1218664906151705051545486763512393729515089
13400452138464531594795202596866463851031515997
1485566498867117803714332959412203163864592721874501485
Technical details
Node id
n-643e863efac6926b · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-d6cf8977dac80cd2

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 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:    1;
  2:    6,    3;
  3:   11,    6;
  4:   32,   39,    9;
  5:   69,  102,   27;
  6:  166,  339,  189,   27;
  7:  367,  894,  630,  108;
  8:  828, 2439, 2349,  810,  81;
  9: 1817, 6174, 7209, 3240, 405;
%F Formula
The row polynomials satisfy (3*n^3-9*n+(12*n^2-13*n)*x+12*n*x^2)*P(n,x) = (15*n^3-9*n^2-36*n+24+(18*n^3+42*n^2-101*n+40)*x+(72*n^2-18*n-42)*x^2+(72*n-72)*x^3)*P(n-1,x) - (18*n^2+36*n+12+(51*n^2+174*n+100)*x+(36*n^2+276*n+283)*x^2+(144*n+336)*x^3+144*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[((15*n^3 - 9*n^2 - 36*n + 24 + (18*n^3 + 42*n^2 - 101*n + 40)*x + (72*n^2 - 18*n - 42)*x^2 + (72*n - 72)*x^3)*P[n - 1, x] - (18*n^2 + 36*n + 12 + (51*n^2 + 174*n + 100)*x + (36*n^2 + 276*n + 283)*x^2 + (144*n + 336)*x^3 + 144*x^4)*D[P[n - 1, x], x])/(3*n^3 - 9*n + (12*n^2 - 13*n)*x + 12*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] = ((15*n^3 - 9*n^2 - 36*n + 24 + (18*n^3 + 42*n^2 - 101*n + 40)*x + (72*n^2 - 18*n - 42)*x^2 + (72*n - 72)*x^3)*v[i - 1] - (18*n^2 + 36*n + 12 + (51*n^2 + 174*n + 100)*x + (36*n^2 + 276*n + 283)*x^2 + (144*n + 336)*x^3 + 144*x^4)*deriv(v[i - 1], x))/(3*n^3 - 9*n + (12*n^2 - 13*n)*x + 12*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. [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.