Real-rooted atlas

A063967 → basis A013610 → D+1

nonnegative coefficients

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

Recurrence

\(\left(10 + 32\,x + 6\,x^{2} - 9\,x^{3} - 7\,n - 47\,n x - 15\,n x^{2} + 9\,n x^{3} - 6\,n^{2} + 15\,n^{2} x + 9\,n^{2} x^{2} + 3\,n^{3}\right)\,P_{n}(x) = \left(-120\,x - 42\,x^{2} + 63\,x^{3} + 27\,x^{4} - 82\,n + 55\,n x + 84\,n x^{2} + 9\,n x^{3} + 48\,n^{2} - 3\,n^{2} x - 18\,n^{2} x^{2} - 6\,n^{3}\right)\,P_{n-1}(x) + \left(-20\,x + 8\,x^{2} + 117\,x^{3} + 108\,x^{4} + 27\,x^{5} - 12\,n + 36\,n x + 147\,n x^{2} + 117\,n x^{3} + 27\,n x^{4} + 12\,n^{2} + 24\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2538
32133963
47620415327460
52609901278621813230
6852422478216561234924322050
72716166084023948627300518505729147475
8848061728185184293760261954127818298892187971000
92606422017678840015522841820151128206851613210278965616314625
1079120761040316893674610721087481710080855585751520054793477331968340656250
1123777625661281218607233413472583113336725146551444072254107537564833116129759049259606250
127086728480832452443681416515042881924923982954683792542312466301771058682962787112838381851771471646212500
13209747227568032163254816574767360133641478821548651942460100896199189554311255395504270030021007375941257743753144110377353125
146171200883756805752884962250119952588783913210857033936144686241091405049303798859179704935827178167705064935836838140920957159432365083625000
15180643202799724801987193808854985585624870722436516416018407874158999589290628235753635169904685124807020943794565644195065512579208471323288094782969406375171875
Technical details
Node id
n-d3c7176cc00d4060 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-3d8aaaa2e018cc33

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:     5,      3;
   3:    21,     33,       9;
   4:    76,    204,     153,      27;
   5:   260,    990,    1278,     621,       81;
   6:   852,   4224,    7821,    6561,     2349,      243;
   7:  2716,  16608,   40239,   48627,    30051,     8505,     729;
   8:  8480,  61728,  185184,  293760,   261954,   127818,   29889,    2187;
   9: 26064, 220176,  788400, 1552284,  1820151,  1282068,  516132,  102789,   6561;
  10: 79120, 761040, 3168936, 7461072, 10874817, 10080855, 5857515, 2005479, 347733, 19683;
%F Formula
The row polynomials satisfy (3*n^3-6*n^2-7*n+10+(15*n^2-47*n+32)*x+(9*n^2-15*n+6)*x^2+(9*n-9)*x^3)*P(n,x) = -(6*n^3-48*n^2+82*n+(3*n^2-55*n+120)*x+(18*n^2-84*n+42)*x^2-(9*n+63)*x^3-27*x^4)*P(n-1,x) + (12*n^2-12*n+(24*n^2+36*n-20)*x+(9*n^2+147*n+8)*x^2+(117*n+117)*x^3+(27*n+108)*x^4+27*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[(-(6*n^3 - 48*n^2 + 82*n + (3*n^2 - 55*n + 120)*x + (18*n^2 - 84*n + 42)*x^2 - (9*n + 63)*x^3 - 27*x^4)*P[n - 1, x] + (12*n^2 - 12*n + (24*n^2 + 36*n - 20)*x + (9*n^2 + 147*n + 8)*x^2 + (117*n + 117)*x^3 + (27*n + 108)*x^4 + 27*x^5)*D[P[n - 1, x], x])/(3*n^3 - 6*n^2 - 7*n + 10 + (15*n^2 - 47*n + 32)*x + (9*n^2 - 15*n + 6)*x^2 + (9*n - 9)*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] = (-(6*n^3 - 48*n^2 + 82*n + (3*n^2 - 55*n + 120)*x + (18*n^2 - 84*n + 42)*x^2 - (9*n + 63)*x^3 - 27*x^4)*v[i - 1] + (12*n^2 - 12*n + (24*n^2 + 36*n - 20)*x + (9*n^2 + 147*n + 8)*x^2 + (117*n + 117)*x^3 + (27*n + 108)*x^4 + 27*x^5)*deriv(v[i - 1], x))/(3*n^3 - 6*n^2 - 7*n + 10 + (15*n^2 - 47*n + 32)*x + (9*n^2 - 15*n + 6)*x^2 + (9*n - 9)*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.