Real-rooted atlas

A099097 → reverse → D+1

nonnegative coefficients

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

Recurrence

\(\left(36 + 15\,x - 12\,x^{2} - 48\,n - 27\,n x + 12\,n x^{2} + 9\,n^{2} + 12\,n^{2} x + 3\,n^{3}\right)\,P_{n}(x) = \left(126 + 65\,x - 32\,x^{2} - 16\,x^{3} - 171\,n - 107\,n x + 18\,n x^{2} + 8\,n x^{3} + 36\,n^{2} + 40\,n^{2} x + 8\,n^{2} x^{2} + 9\,n^{3} + 2\,n^{3} x\right)\,P_{n-1}(x) + \left(63\,x + x^{2} - 48\,x^{3} - 16\,x^{4} - 54\,n x - 60\,n x^{2} - 16\,n x^{3} - 9\,n^{2} x - 4\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
233
310111
433639
5108291138
63511269486
711345135711705
836451998306125961
9116647533148594120777
10371792770267236001572219
11118098998732891733751401250404
1237397735429411955617388103518866268
131180980124002947895383916661519512990689
143720087429089418698853849123817816382110305615
1511691702147032017144929169492520412011718259135450855
Technical details
Node id
n-a31a5cbb16b84035 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-74e1172a7e9b0f1e

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:     3;
   3:    10,     1;
   4:    33,     6;
   5:   108,    29,    1;
   6:   351,   126,    9;
   7:  1134,   513,   57,   1;
   8:  3645,  1998,  306,  12;
   9: 11664,  7533, 1485,  94,  1;
  10: 37179, 27702, 6723, 600, 15;
%F Formula
The row polynomials satisfy (3*n^3+9*n^2-48*n+36+(12*n^2-27*n+15)*x+(12*n-12)*x^2)*P(n,x) = (9*n^3+36*n^2-171*n+126+(2*n^3+40*n^2-107*n+65)*x+(8*n^2+18*n-32)*x^2+(8*n-16)*x^3)*P(n-1,x) - ((9*n^2+54*n-63)*x+(4*n^2+60*n-1)*x^2+(16*n+48)*x^3+16*x^4)*d/dx P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[((9*n^3 + 36*n^2 - 171*n + 126 + (2*n^3 + 40*n^2 - 107*n + 65)*x + (8*n^2 + 18*n - 32)*x^2 + (8*n - 16)*x^3)*P[n - 1, x] - ((9*n^2 + 54*n - 63)*x + (4*n^2 + 60*n - 1)*x^2 + (16*n + 48)*x^3 + 16*x^4)*D[P[n - 1, x], x])/(3*n^3 + 9*n^2 - 48*n + 36 + (12*n^2 - 27*n + 15)*x + (12*n - 12)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = ((9*n^3 + 36*n^2 - 171*n + 126 + (2*n^3 + 40*n^2 - 107*n + 65)*x + (8*n^2 + 18*n - 32)*x^2 + (8*n - 16)*x^3)*v[i - 1] - ((9*n^2 + 54*n - 63)*x + (4*n^2 + 60*n - 1)*x^2 + (16*n + 48)*x^3 + 16*x^4)*deriv(v[i - 1], x))/(3*n^3 + 9*n^2 - 48*n + 36 + (12*n^2 - 27*n + 15)*x + (12*n - 12)*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. [MW08] Shi-Mei Ma and Yi Wang. q-Eulerian polynomials and polynomials with only real zeros. The Electronic Journal of Combinatorics, 15(1):Research Paper 17, January 2008. doi:10.37236/741
  3. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.