Real-rooted atlas

A153651 → basis A008277

nonnegative coefficients

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

Recurrence

\(\left(-2 + n\right)\,P_{n}(x) = \left(-286 - 286\,x + 143\,n + 143\,n x\right)\,P_{n-2}(x) + \left(-286\,x + 143\,n x\right)\,P_{n-2}'(x) + \left(-4 - 2\,x + 2\,n + n x\right)\,P_{n-1}(x) + \left(-2\,x + n x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 2\)
\(P_{1}(x) = 26 + 13\,x\)
\(P_{2}(x) = 338 + 340\,x + 2\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
022
1261339
23383402680
3439487942207215397
4571221763506710025032303077
574258637295982649095396360280127520442
6965361870535926683299281565791177252531012164953011
712549703413941162222013649907752157988789752561196491340324365596303
816314614422586158791048848892380245019446434017446172200482426166916037072105063487842
9212089987464958795405261284471587279889544921432212632521999177408740403883403222191440401322921868979799
1027571698369891385607487903028814779179226732700140575854159215035510833273643835118741269465099397027642454321276111886354825
1135843207880741725458346411827522402371538678564507805346683572194300367666175300427321344185003159771150860318309971770713388495463122208328194345776
12465961702449623174277727534374174819363730123882465680375446236312929781868361314294043301609063630325542001449213518811719369221707523565143591658140651164943261546826288950105
13605750213184506594300959329207184204871393122818957382261555241572804801198490573754611375792691426700581862007480356900811884286910870443347655639708638494447495919765373214795040525721856290626159455635
1478747527713985781093402720431807190969505107224177823220715226565826311415163896622226348674655815025068037499457101058046430668956288146978692382896374172191375582671468482202260620393421362262866661755792055573254818783658366365640
Technical details
Node id
n-a680e08696d5bf64 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-81fde51522ebbafc

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:           2;
  1:          26,           13;
  2:         338,          340,             2;
  3:        4394,         8794,          2207,            2;
  4:       57122,       176350,         67100,         2503,            2;
  5:      742586,      3729598,       2649095,       396360,         2801,           2;
  6:     9653618,     70535926,      68329928,     15657911,       772525,        3101,         2;
  7:   125497034,   1394116222,    2013649907,    752157988,     78975256,     1196491,      3403,       2;
  8:  1631461442,  25861587910,   48848892380,  24501944643,   4017446172,   200482426,   1669160,    3707,    2;
  9: 21208998746, 495879540526, 1284471587279, 889544921432, 212632521999, 17740874040, 388340322, 2191440, 4013, 2;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = (143*n-286+(143*n-286)*x)*P(n-2,x) + (143*n-286)*x*d/dx P(n-2,x) + (2*n-4+(n-2)*x)*P(n-1,x) + (n-2)*x*d/dx P(n-1,x) for n >= 3, with P(0,x) = 2, P(1,x) = 26 + 13*x, P(2,x) = 338 + 340*x + 2*x^2.
%t Mathematica
P[0, x_] := 2;
P[1, x_] := 26 + 13*x;
P[2, x_] := 338 + 340*x + 2*x^2;
P[n_, x_] := P[n, x] = Expand[Together[((143*n - 286 + (143*n - 286)*x)*P[n - 2, x] + (143*n - 286)*x*D[P[n - 2, x], x] + (2*n - 4 + (n - 2)*x)*P[n - 1, x] + (n - 2)*x*D[P[n - 1, x], x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 2; v[2] = 26 + 13*x; v[3] = 338 + 340*x + 2*x^2; for(n = 3, N, my(i = n + 1); v[i] = ((143*n - 286 + (143*n - 286)*x)*v[i - 2] + (143*n - 286)*x*deriv(v[i - 2], x) + (2*n - 4 + (n - 2)*x)*v[i - 1] + (n - 2)*x*deriv(v[i - 1], x))/(n - 2)); 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

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