Real-rooted atlas

A269951 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(3 - n\right)\,P_{n-2}(x) + \left(x + n\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)
\(P_{2}(x) = 1 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2112
33519
4112711150
549163102191334
626111099462702912616
7163184819215357058541123524
811743724159590147135104451113551238808
99590168500710720366401011794102578119327112700240
108768097124073128781329063404307872655833456308313289133639008
118877691808856291659635011346449805380901411805822150011411211648151091457603792
129864101199627421122884714552104067565967880670249482982383011023602130207615709513116746935968
1311935562331323762367533663711026346071026951801994881553405161269622027801092225427913895362615100981551107142170656
1415624736141188798895037526683814086598839713626348319493995116745442407241632596043216252732281044467161759956035261396218111822689445760
152200483673192877555572633873845920218710890513427014700067071375126191421486236126274607319361014920495853087336649169673114296996467818837209133062909393856
Technical details
Node id
n-6772d94cb42ddb72 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-52b1d98f66e789d0

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:      1,       1;
   3:      3,       5,        1;
   4:     11,      27,       11,       1;
   5:     49,     163,      102,      19,       1;
   6:    261,    1109,      946,     270,      29,      1;
   7:   1631,    8481,     9215,    3570,     585,     41,     1;
   8:  11743,   72415,    95901,   47135,   10445,   1113,    55,    1;
   9:  95901,  685007,  1072036,  640101,  179410,  25781,  1932,   71,  1;
  10: 876809, 7124073, 12878132, 9063404, 3078726, 558334, 56308, 3132, 89, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(n-3)*P(n-2,x) + (n+x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 1, P(2,x) = 1 + x.
%t Mathematica
P[1, x_] := 1;
P[2, x_] := 1 + x;
P[n_, x_] := P[n, x] = Expand[-(n - 3)*P[n - 2, x] + (n + x)*P[n - 1, x] + x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; v[2] = 1 + x; for(n = 3, N, my(i = n); v[i] = -(n - 3)*v[i - 2] + (n + x)*v[i - 1] + x*deriv(v[i - 1], x)); 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

  • Real-rooted? Every row P_n, n ≥ 1, is zero or real-rooted with all roots in (−∞, 0]
    by Ray basis map theorem
  • Coefficients of one sign? Every nonzero row, n ≥ 1: the coefficients of P_n have one sign
    derived from Ray basis map theorem
  • Hadamard kernel (Malo)? Rows P_n, n ≥ 1, are Hadamard kernels (preserve real-rootedness)
    derived from Ray basis map theorem

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