Real-rooted atlas

A113214 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(-21\,n - 252\,n x - 54\,n x^{2} + 81\,n x^{3} - 63\,n^{2} + 81\,n^{2} x + 81\,n^{2} x^{2} + 27\,n^{3}\right)\,P_{n}(x) = \left(75 - 1350\,x - 702\,x^{2} + 486\,x^{3} + 243\,x^{4} - 786\,n + 72\,n x + 513\,n x^{2} + 81\,n x^{3} + 423\,n^{2} + 81\,n^{2} x - 162\,n^{2} x^{2} - 54\,n^{3}\right)\,P_{n-1}(x) + \left(10 - 159\,x - 504\,x^{2} + 432\,x^{3} + 810\,x^{4} + 243\,x^{5} - 105\,n - 243\,n x + 432\,n x^{2} + 729\,n x^{3} + 243\,n x^{4} + 45\,n^{2} + 162\,n^{2} x + 81\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1639
21933961
34616514427382
498555963567812264
5198153041584725210624312960
63833774141032486720979753372972368
772286494132810025113008687237262442187396704

All 15 rows

Technical details
Node id
n-72f0e68748df8a47 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-32139e209585c36c

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 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:    6,     3;
  2:   19,    33,      9;
  3:   46,   165,    144,      27;
  4:   98,   555,    963,     567,      81;
  5:  198,  1530,   4158,    4725,    2106,     243;
  6:  383,  3774,  14103,   24867,   20979,    7533,     729;
  7:  722,  8649,  41328,  100251,  130086,   87237,   26244,    2187;
  8: 1333, 18825, 109791,  341145,  607824,  622323,  346275,   89667,   6561;
  9: 2424, 39435, 271854, 1033101, 2358558, 3306744, 2792070, 1327509, 301806, 19683;
%F Formula
The row polynomials satisfy (27*n^3-63*n^2-21*n+(81*n^2-252*n)*x+(81*n^2-54*n)*x^2+81*n*x^3)*P(n,x) = -(54*n^3-423*n^2+786*n-75-(81*n^2+72*n-1350)*x+(162*n^2-513*n+702)*x^2-(81*n+486)*x^3-243*x^4)*P(n-1,x) + (45*n^2-105*n+10+(162*n^2-243*n-159)*x+(81*n^2+432*n-504)*x^2+(729*n+432)*x^3+(243*n+810)*x^4+243*x^5)*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[(-(54*n^3 - 423*n^2 + 786*n - 75 - (81*n^2 + 72*n - 1350)*x + (162*n^2 - 513*n + 702)*x^2 - (81*n + 486)*x^3 - 243*x^4)*P[n - 1, x] + (45*n^2 - 105*n + 10 + (162*n^2 - 243*n - 159)*x + (81*n^2 + 432*n - 504)*x^2 + (729*n + 432)*x^3 + (243*n + 810)*x^4 + 243*x^5)*D[P[n - 1, x], x])/(27*n^3 - 63*n^2 - 21*n + (81*n^2 - 252*n)*x + (81*n^2 - 54*n)*x^2 + 81*n*x^3)]];
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] = (-(54*n^3 - 423*n^2 + 786*n - 75 - (81*n^2 + 72*n - 1350)*x + (162*n^2 - 513*n + 702)*x^2 - (81*n + 486)*x^3 - 243*x^4)*v[i - 1] + (45*n^2 - 105*n + 10 + (162*n^2 - 243*n - 159)*x + (81*n^2 + 432*n - 504)*x^2 + (729*n + 432)*x^3 + (243*n + 810)*x^4 + 243*x^5)*deriv(v[i - 1], x))/(27*n^3 - 63*n^2 - 21*n + (81*n^2 - 252*n)*x + (81*n^2 - 54*n)*x^2 + 81*n*x^3)); 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. [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.