Real-rooted atlas

A075504 → D

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-162 - 18\,x - x^{2}\right)\,P_{n-2}(x) + \left(-81\,x - 9\,x^{2}\right)\,P_{n-2}'(x) + \left(27 + 2\,x\right)\,P_{n-1}(x) + \left(9\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 27 + 2\,x\)
\(P_{1}(x) = 567 + 108\,x + 3\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15Pn(1)
027229
15671083678
2109354050270415259
3203391131220157955405350951
4372008739497227654504536094568481570
56749300711408266833480783306180010773015127218227507
61219657095321521805013764321901824220449644670224532226886003600857
7219968744318925020154054374385915100442349007488397352571912042525032409176440684140
839633116024724537491904422091353195250524489751720529926393152518912242606236407484404455105521496225761
97137447668847670438904624287896933881466326390509754400351721471266022228263842473036330291299078001242945594011183288118841147
10128505439098855182445993782325029434620330215101292865390700884222917731676640182244638759047851913229161886954097625801132519691107722126431239317049104
1123133803333158714951742919942546010870132309550923562103558625568460136536392489146651415642502193920777956497718402425794330960444492528484816211402999997982813237785239382914237
12416433878655140071341597349051119162398752377523747065029395952356112596808149259677233524851055698447369145587252869702798720285398654832720672945137865097068551580853782930442260012285149238400655961410041
13749603858371707087363064152001112453881455632519321256221431375984839694899798004768049630157510879076288113796936401366453714374612414913140628498750600092672411076531017086954804920198983963178144884376063390601512015376204784375320782198
141349307534182282221598177241811974703965052947366923566892652306386016114536549920524274674490524691834941053783171248258945911655367023440185300844328147644108654770411873724057982215027213019820929804028591830863638658813429623683023008867880183601616019364917183034251311
Technical details
Node id
n-7984a7701e96ffec · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-a1e10db0ebddaa14

Observed on the stored rows ?

  • Degrees: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 (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 = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:            27,              2;
  1:           567,            108,              3;
  2:         10935,           4050,            270,              4;
  3:        203391,         131220,          15795,            540,             5;
  4:       3720087,        3949722,         765450,          45360,           945,            6;
  5:      67493007,      114082668,       33480783,        3061800,        107730,         1512,          7;
  6:    1219657095,     3215218050,     1376432190,      182422044,       9644670,       224532,       2268,         8;
  7:   21996874431,    89250201540,    54374385915,    10044234900,     748839735,     25719120,     425250,      3240,       9;
  8:  396331160247,  2453749190442,  2091353195250,   524489751720,   52992639315,   2518912242,   60623640,    748440,    4455,   10;
  9: 7137447668847, 67043890462428, 78969338814663, 26390509754400, 3517214712660, 222282638424, 7303633029, 129907800, 1242945, 5940, 11;
%F Formula
The row polynomials satisfy P(n,x) = -(162+18*x+x^2)*P(n-2,x) - (81*x+9*x^2)*d/dx P(n-2,x) + (27+2*x)*P(n-1,x) + 9*x*d/dx P(n-1,x) for n >= 2, with P(0,x) = 27 + 2*x, P(1,x) = 567 + 108*x + 3*x^2.
%t Mathematica
P[0, x_] := 27 + 2*x;
P[1, x_] := 567 + 108*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[-(162 + 18*x + x^2)*P[n - 2, x] - (81*x + 9*x^2)*D[P[n - 2, x], x] + (27 + 2*x)*P[n - 1, x] + 9*x*D[P[n - 1, x], x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 27 + 2*x; v[2] = 567 + 108*x + 3*x^2; for(n = 2, N, my(i = n + 1); v[i] = -(162 + 18*x + x^2)*v[i - 2] - (81*x + 9*x^2)*deriv(v[i - 2], x) + (27 + 2*x)*v[i - 1] + 9*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 - 1, v[i])))};
T(9)

polytool c48629374597

Properties

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. [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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.