Real-rooted atlas

A143495 → D+1 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-20 - 8\,x - x^{2}\right)\,P_{n-2}(x) + \left(-4\,x - x^{2}\right)\,P_{n-2}'(x) + \left(9 + 2\,x\right)\,P_{n-1}(x) + \left(x\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 0\)
\(P_{2}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13Pn(1)
10
211
39211
46130394
5369302664741
62101255091512055691
71152919502102302180195643642
8617411400701008633108044802947338535
932508996390291614638312479170831642082676175
101690981643359078588754286520119143517816414322576921654472
1187170494198350264662510447706601608106532113623654002328076510179815603
124463382126933511051561282344438196020069291051410898776598969696036135990111534567607
1322736340917052783024013292426424253944423617136407549404721454052601723893612538355401012541213469081000
1411535942611068639663030650964435392913721202655859706510383461088248022174437304467235693658214929078221156013121615574757
155835080169664255683022305863459903551989295402881415887351357049273583939647712073110752008836928106974968035375341102921911141129557084655
Technical details
Node id
n-c656007927f23726 · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-827081d473d1a303

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:       1;
   3:       9,       2;
   4:      61,      30,       3;
   5:     369,     302,      66,       4;
   6:    2101,    2550,     915,     120,       5;
   7:   11529,   19502,   10230,    2180,     195,      6;
   8:   61741,  140070,  100863,   31080,    4480,    294,     7;
   9:  325089,  963902,  916146,  383124,   79170,   8316,   420,   8;
  10: 1690981, 6433590, 7858875, 4286520, 1191435, 178164, 14322, 576, 9;
%F Formula
The row polynomials satisfy P(n,x) = -(20+8*x+x^2)*P(n-2,x) - (4*x+x^2)*d/dx P(n-2,x) + (9+2*x)*P(n-1,x) + x*d/dx P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 1.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 1;
P[n_, x_] := P[n, x] = Expand[-(20 + 8*x + x^2)*P[n - 2, x] - (4*x + x^2)*D[P[n - 2, x], x] + (9 + 2*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] = 0; v[2] = 1; for(n = 3, N, my(i = n); v[i] = -(20 + 8*x + x^2)*v[i - 2] - (4*x + x^2)*deriv(v[i - 2], x) + (9 + 2*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

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  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. [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
  5. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.