Real-rooted atlas

A104684 → basis A013611 → D

nonnegative coefficients

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

Recurrence

\(\left(8 - 8\,n + 2\,n^{2}\right)\,P_{n}(x) = \left(46 + 88\,x - 52\,n - 96\,n x + 14\,n^{2} + 24\,n^{2} x\right)\,P_{n-1}(x) + \left(3 + 18\,x + 24\,x^{2} - 2\,n - 12\,n x - 16\,n x^{2}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 4\)

Rows ?

nx0x1x2x3x4x5x6Pn(1)
10
244
3323264
4228480192900
5153649924608102412160
6100204416065280358405120160420
764032356640729600660480245760245762081088
84033962716896711648090675205698560154828811468826665828

All 15 rows

Technical details
Node id
n-4a95cea6638bf316 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-b91fbc6aa50ad79c

Observed on the stored rows ?

  • Degrees: -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13 (affine: slope 1, intercept -2)
  • 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:        0;
   2:        4;
   3:       32,        32;
   4:      228,       480,       192;
   5:     1536,      4992,      4608,       1024;
   6:    10020,     44160,     65280,      35840,       5120;
   7:    64032,    356640,    729600,     660480,     245760,     24576;
   8:   403396,   2716896,   7116480,    9067520,    5698560,   1548288,    114688;
   9:  2514944,  19873280,  63522816,  104796160,   94044160,  44040192,   9175040,   524288;
  10: 15554628, 141083136, 532675584, 1080963072, 1268305920, 858783744, 313786368, 51904512, 2359296;
%F Formula
The row polynomials satisfy (2*n^2-8*n+8)*P(n,x) = (14*n^2-52*n+46+(24*n^2-96*n+88)*x)*P(n-1,x) - (2*n-3+(12*n-18)*x+(16*n-24)*x^2)*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 4.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 4;
P[n_, x_] := P[n, x] = Expand[Together[((14*n^2 - 52*n + 46 + (24*n^2 - 96*n + 88)*x)*P[n - 1, x] - (2*n - 3 + (12*n - 18)*x + (16*n - 24)*x^2)*D[P[n - 1, x], x])/(2*n^2 - 8*n + 8)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 4; for(n = 3, N, my(i = n); v[i] = ((14*n^2 - 52*n + 46 + (24*n^2 - 96*n + 88)*x)*v[i - 1] - (2*n - 3 + (12*n - 18)*x + (16*n - 24)*x^2)*deriv(v[i - 1], x))/(2*n^2 - 8*n + 8)); 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

Same fingerprint ?

References

  1. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  2. [Fav35] Jean Favard. Sur les polynômes de tchebicheff. C. R. Acad. Sci. Paris, 200:2052–2053, 1935.
  3. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.
  5. [Sze39] G. Szegő. Orthogonal Polynomials. Colloquium publications, American Mathematical Society, 1939. doi:10.1090/coll/023