Real-rooted atlas

A272867 → basis A013610 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(33 + 45\,x + 9\,x^{2}\right)\,P_{n-1}(x) + \left(45\,x + 18\,x^{2}\right)\,P_{n-1}'(x) + \left(9\,x^{2}\right)\,P_{n-1}''(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
13345987
210894995440111348111700
335937438615945027653184177876196837292271051
41185921358291351612049852255015701302686553531130246758062886846561594272619
5391353932848038975249978116796344141454065591738550327372970418618680503124278462097332435383818559049199618130970
612914679692239081327353704470120701161344491246542756062841912122584142245168994836467570425155554244483773914887423374916070517468465674782969053144183085745467645
7426184429771752295041193553644812259286738839249586445841047761241696489613261657870542627899731781804816135516503137384728582154250635181306185787179001264127227574476938839334028280101374569173311478296941728450587894675

All 15 rows

Technical details
Node id
n-2a75e568b1aee671 · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-63c8a826cb5a9f43

Observed on the stored rows ?

  • Degrees: 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 (affine: slope 2, 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:             33,                 45,                    9;
  2:           1089,               4995,                 4401,                  1134,                     81;
  3:          35937,             438615,               945027,                653184,                 177876,                  19683,                    729;
  4:        1185921,           35829135,            161204985,             225501570,              130268655,               35311302,                4675806,                 288684,                   6561;
  5:       39135393,         2848038975,          24997811679,           63441414540,            65591738550,            32737297041,             8618680503,             1242784620,               97332435,               3838185,                 59049;
  6:     1291467969,       223908132735,        3704470120701,        16134449124654,         27560628419121,         22584142245168,          9948364675704,          2515555424448,           377391488742,           33749160705,            1746846567,             47829690,              531441;
  7:    42618442977,     17522950411935,      536448122592867,      3883924958644584,      10477612416964896,      13261657870542627,       8997317818048161,       3551650313738472,        858215425063518,       130618578717900,        12641272275744,         769388393340,         28280101374,          569173311,          4782969;
  8:  1406408618241,   1368707962064895,    76743514727532945,    905841345302827290,    3752650189854881235,    7069442124710364714,    7040085434017534962,    4074442674770382948,    1458127300293687807,    336019344648290550,     51107149672713954,     5187785051593620,     350534088582930,     15456470433516,     424325877804,      6543101592,      43046721;
  9: 46411484401953, 106822509428882655, 10907076835869797799, 207325321721816410980, 1294969695760944574650, 3543628294943886294837, 5022117492236640700731, 4098803432410374151860, 2068030654655331261645, 677173052116483979355, 148760694571658476635, 22398340486514369760, 2337886644586214976, 169432445801056662, 8443535893024338, 282131997545448, 6005404999989, 73222472421, 387420489;
%F Formula
The row polynomials satisfy P(n,x) = (33+45*x+9*x^2)*P(n-1,x) + (45*x+18*x^2)*d/dx P(n-1,x) + 9*x^2*d^2/dx^2 P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(33 + 45*x + 9*x^2)*P[n - 1, x] + (45*x + 18*x^2)*D[P[n - 1, x], x] + 9*x^2*D[P[n - 1, x], {x, 2}]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (33 + 45*x + 9*x^2)*v[i - 1] + (45*x + 18*x^2)*deriv(v[i - 1], x) + 9*x^2*deriv(deriv(v[i - 1], x), x)); 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. [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. [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
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.