Real-rooted atlas

A026729 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(3 + 225\,x + 135\,x^{2} - 81\,x^{3} + 60\,n - 306\,n x - 216\,n x^{2} + 81\,n x^{3} - 90\,n^{2} + 81\,n^{2} x + 81\,n^{2} x^{2} + 27\,n^{3}\right)\,P_{n}(x) = \left(300 - 675\,x - 783\,x^{2} + 243\,x^{3} + 243\,x^{4} - 822\,n + 18\,n x + 675\,n x^{2} + 81\,n x^{3} + 423\,n^{2} + 81\,n^{2} x - 162\,n^{2} x^{2} - 54\,n^{3}\right)\,P_{n-1}(x) + \left(40 - 6\,x - 423\,x^{2} + 27\,x^{3} + 567\,x^{4} + 243\,x^{5} - 105\,n - 243\,n x + 432\,n x^{2} + 729\,n x^{3} + 243\,n x^{4} + 45\,n^{2} + 162\,n^{2} x + 81\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
111
2437
31127947
42411112627288
550333711513811688
6988642736369919442439584
7187204686311746917091704772953200
8348455724066653139590473143247862187290304

All 15 rows

Technical details
Node id
n-cc2d6f4d77b24cb6 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-4ff62ab1125a6851

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 -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:    4,     3;
   3:   11,    27,      9;
   4:   24,   111,    126,     27;
   5:   50,   333,    711,    513,      81;
   6:   98,   864,   2736,   3699,    1944,     243;
   7:  187,  2046,   8631,  17469,   17091,    7047,     729;
   8:  348,  4557,  24066,  65313,   95904,   73143,   24786,    2187;
   9:  637,  9711,  61659, 210519,  416016,  476037,  296703,   85293,   6561;
  10: 1150, 20013, 148536, 612063, 1526526, 2354670, 2198664, 1156923, 288684, 19683;
%F Formula
The row polynomials satisfy (27*n^3-90*n^2+60*n+3+(81*n^2-306*n+225)*x+(81*n^2-216*n+135)*x^2+(81*n-81)*x^3)*P(n,x) = -(54*n^3-423*n^2+822*n-300-(81*n^2+18*n-675)*x+(162*n^2-675*n+783)*x^2-(81*n+243)*x^3-243*x^4)*P(n-1,x) + (45*n^2-105*n+40+(162*n^2-243*n-6)*x+(81*n^2+432*n-423)*x^2+(729*n+27)*x^3+(243*n+567)*x^4+243*x^5)*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[(-(54*n^3 - 423*n^2 + 822*n - 300 - (81*n^2 + 18*n - 675)*x + (162*n^2 - 675*n + 783)*x^2 - (81*n + 243)*x^3 - 243*x^4)*P[n - 1, x] + (45*n^2 - 105*n + 40 + (162*n^2 - 243*n - 6)*x + (81*n^2 + 432*n - 423)*x^2 + (729*n + 27)*x^3 + (243*n + 567)*x^4 + 243*x^5)*D[P[n - 1, x], x])/(27*n^3 - 90*n^2 + 60*n + 3 + (81*n^2 - 306*n + 225)*x + (81*n^2 - 216*n + 135)*x^2 + (81*n - 81)*x^3)]];
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] = (-(54*n^3 - 423*n^2 + 822*n - 300 - (81*n^2 + 18*n - 675)*x + (162*n^2 - 675*n + 783)*x^2 - (81*n + 243)*x^3 - 243*x^4)*v[i - 1] + (45*n^2 - 105*n + 40 + (162*n^2 - 243*n - 6)*x + (81*n^2 + 432*n - 423)*x^2 + (729*n + 27)*x^3 + (243*n + 567)*x^4 + 243*x^5)*deriv(v[i - 1], x))/(27*n^3 - 90*n^2 + 60*n + 3 + (81*n^2 - 306*n + 225)*x + (81*n^2 - 216*n + 135)*x^2 + (81*n - 81)*x^3)); 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.