Real-rooted atlas

A146987 → basis A007318

nonnegative coefficients

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

Recurrence

\(\left(-3 + n\right)\,P_{n}(x) = \left(36 + 36\,x + 9\,x^{2} - 12\,n - 12\,n x - 3\,n x^{2}\right)\,P_{n-2}(x) + \left(-24 - 12\,x + 8\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 2 + x\)
\(P_{2}(x) = 7 + 7\,x + x^{2}\)
\(P_{3}(x) = 26 + 39\,x + 15\,x^{2} + x^{3}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1213
277115
3263915181
4124248159351567
5680170015386079114617
63952118561384879362247255140095
72345682096117312880403673081037431356481
814022456089694657687659248226015791228543220313195207
984032037814407352928808852855152722390220643650985596579128717497
1050398722519936055438848705592325733033630871008110258402520384334791197031258339375
113023513616629324840817484858958745655557523235715792015873849648189264956725811221515907112324700081
12181402624108841574429477936644761824000510196032038094704642023786464765300384201955500354311603720351177171120921237847
13108839936070745958402095168972837413739264448965017603816204480023570690880106629420483507551784818430580128611010122234555314671188287951977
146530363392457125437441469331804162873360138243816181783043636597240322559087421441346888463365303375246415433110704322692576845916793639858439159435111694582001855
Technical details
Node id
n-6587bc79a3c1f503 · revision 0 · JSON
Definition source
derived:basis_map_A007318
Fingerprint
f6-d3587b040c864a47

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 0)
  • 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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:      1;
  1:      2,       1;
  2:      7,       7,       1;
  3:     26,      39,      15,       1;
  4:    124,     248,     159,      35,       1;
  5:    680,    1700,    1538,     607,      91,       1;
  6:   3952,   11856,   13848,    7936,    2247,     255,      1;
  7:  23456,   82096,  117312,   88040,   36730,    8103,    743,     1;
  8: 140224,  560896,  946576,  876592,  482260,  157912,  28543,  2203,    1;
  9: 840320, 3781440, 7352928, 8088528, 5515272, 2390220, 643650, 98559, 6579, 1;
%F Formula
The row polynomials satisfy (n-3)*P(n,x) = -(12*n-36+(12*n-36)*x+(3*n-9)*x^2)*P(n-2,x) + (8*n-24+(4*n-12)*x)*P(n-1,x) for n >= 4, with P(0,x) = 1, P(1,x) = 2 + x, P(2,x) = 7 + 7*x + x^2, P(3,x) = 26 + 39*x + 15*x^2 + x^3.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 2 + x;
P[2, x_] := 7 + 7*x + x^2;
P[3, x_] := 26 + 39*x + 15*x^2 + x^3;
P[n_, x_] := P[n, x] = Expand[Together[(-(12*n - 36 + (12*n - 36)*x + (3*n - 9)*x^2)*P[n - 2, x] + (8*n - 24 + (4*n - 12)*x)*P[n - 1, x])/(n - 3)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 2 + x; v[3] = 7 + 7*x + x^2; v[4] = 26 + 39*x + 15*x^2 + x^3; for(n = 4, N, my(i = n + 1); v[i] = (-(12*n - 36 + (12*n - 36)*x + (3*n - 9)*x^2)*v[i - 2] + (8*n - 24 + (4*n - 12)*x)*v[i - 1])/(n - 3)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

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