Real-rooted atlas

A038226 → basis A013609 → basis A013610

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(27 + 48\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 729 + 2592\,x + 2304\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
0729259223045625
119683104976186624110592421875
253144137791361007769611943936530841631640625
3143489071275458404534963208062156807166361602548039682373046875
438742048941324852161836660096043535646720580475289604127824281612230590464177978515625
510460353203130173284304694257516288205705930752036569943244803900793946112231158159769658706834227213348388671875
628242953648140167756299522499327058636888864962084864197477693521920280857164120064249650812551168126806761930752281792804290561001129150390625
77625597484987122009559759792867623536069632359903096443699295974159051653121706207272029388820221715816644608154070215745863686847565144260608135260546059468875084686279296875

All 15 rows

Technical details
Node id
n-824192ee04ae06b8 · revision 0 · JSON
Definition source
derived:basis_map_A013610
Fingerprint
f6-f3c17ca3ebbc301e

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): yes

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:              729,               2592,               2304;
  1:            19683,             104976,             186624,              110592;
  2:           531441,            3779136,           10077696,            11943936,              5308416;
  3:         14348907,          127545840,          453496320,           806215680,            716636160,            254803968;
  4:        387420489,         4132485216,        18366600960,         43535646720,          58047528960,          41278242816,          12230590464;
  5:      10460353203,       130173284304,       694257516288,       2057059307520,        3656994324480,        3900793946112,        2311581597696,          587068342272;
  6:     282429536481,      4016775629952,     24993270586368,      88864962084864,      197477693521920,      280857164120064,      249650812551168,       126806761930752,       28179280429056;
  7:    7625597484987,    122009559759792,    867623536069632,    3599030964436992,     9597415905165312,    17062072720293888,    20221715816644608,     15407021574586368,     6847565144260608,     1352605460594688;
  8:  205891132094649,   3660286792793760,  29282294342350080,  138819765771141120,   431883715732439040,   921351926895869952,  1364965817623511040,   1386631941712773120,   924421294475182080,   365203474360565760,    64925062108545024;
  9: 5559060566555523, 108710517745974672, 966315713297552640, 5153683804253614080, 18324209081790627840, 45606920381345562624, 81078969566836555776, 102957421672173404160, 91517708153043025920, 54232715942544015360, 19282743446237872128, 3116402981210161152;
%F Formula
The row polynomials satisfy P(n,x) = (27+48*x)*P(n-1,x) for n >= 1, with P(0,x) = 729 + 2592*x + 2304*x^2.
%t Mathematica
P[0, x_] := 729 + 2592*x + 2304*x^2;
P[n_, x_] := P[n, x] = Expand[(27 + 48*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 729 + 2592*x + 2304*x^2; for(n = 1, N, my(i = n + 1); v[i] = (27 + 48*x)*v[i - 1]); 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

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