Real-rooted atlas

A038329 → basis A013611 → basis A008277

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(27 + 12\,x\right)\,P_{n-1}(x) + \left(12\,x\right)\,P_{n-1}'(x)\)
\(P_{0}(x) = 729 + 792\,x + 144\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
07297921441665
1196833963616848172877895
253144117820001334880311040207363980097
3143489077587529289462880356140805287680248832220837671
438742048931313232725473110384331724160082394496085100544298598413221127233
51046035320312677065347631670450884827466354540810160277120017291086848131681894435831808848845569735
6282429536481506957972400017673177792960211042574668801091617038489627235578101763378581176321977915801642998169658127239972737
776255974849872011027636737729621670241289601541646353928960107196386846976036794357410406466130647367680624978395136029023764480051597803524225125010534695

All 15 rows

Technical details
Node id
n-47b7c605b3a36a4e · revision 0 · JSON
Definition source
derived:basis_map_A008277
Fingerprint
f6-65e248b1aaac025a

Observed on the stored rows ?

  • Degrees: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 (affine: slope 1, intercept 2)
  • 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:              729,                792,                 144;
  1:            19683,              39636,               16848,                1728;
  2:           531441,            1782000,             1334880,              311040,               20736;
  3:         14348907,           75875292,            89462880,            35614080,             5287680,              248832;
  4:        387420489,         3131323272,          5473110384,          3317241600,           823944960,            85100544,             2985984;
  5:      10460353203,       126770653476,        316704508848,        274663545408,        101602771200,         17291086848,          1316818944,           35831808;
  6:     282429536481,      5069579724000,      17673177792960,      21104257466880,      10916170384896,       2723557810176,        337858117632,        19779158016,         429981696;
  7:    7625597484987,    201102763673772,     962167024128960,    1541646353928960,    1071963868469760,     367943574104064,      66130647367680,      6249783951360,      290237644800,       5159780352;
  8:  205891132094649,   7934514953096952,   51483751394662224,  108669724587072000,   98897046382379520,   44874657368690688,   10962256978649088,   1487293787013120,   110696637726720,    4179422085120,    61917364224;
  9: 5559060566555523, 311916776755916916, 2720885500564936848, 7463997665721482688, 8721315173723328000, 5090859747664644096, 1623759329310547968, 296636694102245376, 31463211884544000, 1892581634211840, 59254917562368, 743008370688;
%F Formula
The row polynomials satisfy P(n,x) = (27+12*x)*P(n-1,x) + 12*x*d/dx P(n-1,x) for n >= 1, with P(0,x) = 729 + 792*x + 144*x^2.
%t Mathematica
P[0, x_] := 729 + 792*x + 144*x^2;
P[n_, x_] := P[n, x] = Expand[(27 + 12*x)*P[n - 1, x] + 12*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] = 729 + 792*x + 144*x^2; for(n = 1, N, my(i = n + 1); v[i] = (27 + 12*x)*v[i - 1] + 12*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

  • 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

Same fingerprint ?

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