Real-rooted atlas

A099097 → basis A007318 → reverse

nonnegative coefficients

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

Recurrence

\(\left(-3 + 3\,n\right)\,P_{n}(x) = \left(-9 - 31\,x - 26\,x^{2} + 9\,n + 20\,n x + 13\,n x^{2}\right)\,P_{n-1}(x) + \left(-9\,x - 22\,x^{2} - 13\,x^{3}\right)\,P_{n-1}'(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
111
2336
39191038
427879333240
5813515684071091516
624313232871310516743609576
772947791301418847153126618118960488
8218716767549459977710845370569254553927382080
965615759122064448195965655157122531000295953129702413456
1019683194643853659217952135702523891558282270313138893561514283715244896
115904964953932090589377775179519222352394021370559132910035416075130588714148196296288
12177147214544711790117388088018502227113017735914214530711069894760257826218359264741092467280608267520
13531441702683142515280155655351384080049672928587858443860803427261547531881264988334864530161707248415433213842197696
14159432322851963150948927608447457166980511832948569804809316971525793602943057216802608517418113641780233715351861055787509724324269721216
15478296973870299528960942232773126370329283561543372169425369530489317331314473035284424822093676436120473258754772155310129815341521707489516835050153302722688
Technical details
Node id
n-90e0961665129420 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-50f0520d3a512303

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:     3,      3;
   3:     9,     19,     10;
   4:    27,     87,     93,      33;
   5:    81,    351,    568,     407,     109;
   6:   243,   1323,   2871,    3105,    1674,     360;
   7:   729,   4779,  13014,   18847,   15312,    6618,    1189;
   8:  2187,  16767,  54945,   99777,  108453,   70569,   25455,    3927;
   9:  6561,  57591, 220644,  481959,  656551,  571225,  310002,   95953,  12970;
  10: 19683, 194643, 853659, 2179521, 3570252, 3891558, 2822703, 1313889, 356151, 42837;
%F Formula
The row polynomials satisfy (3*n-3)*P(n,x) = (9*n-9+(20*n-31)*x+(13*n-26)*x^2)*P(n-1,x) - (9*x+22*x^2+13*x^3)*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[((9*n - 9 + (20*n - 31)*x + (13*n - 26)*x^2)*P[n - 1, x] - (9*x + 22*x^2 + 13*x^3)*D[P[n - 1, x], x])/(3*n - 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] = ((9*n - 9 + (20*n - 31)*x + (13*n - 26)*x^2)*v[i - 1] - (9*x + 22*x^2 + 13*x^3)*deriv(v[i - 1], x))/(3*n - 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

Same fingerprint ?

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