Real-rooted atlas

A038321 → basis A007318 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(7 + 18\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 49 + 252\,x + 324\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
049252324625
134326466804583215625
224012469695256163296104976390625
31680721609011113202857680367416018895689765625
411764918151561166886040007520771573607936185634012224244140625
5823543148237741143548284900921201260236880194436547216665989766122200326103515625
65764801118590192106731172854890317441764331632036294822144466647713283428432179211019960576152587890625
74035360793389776296058055525763483331222230578563257164344876897996019788810799561364486942575162881983592903683814697265625

All 15 rows

Technical details
Node id
n-90ff96431b809a39 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-c53984b4362c13dd

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:         49,         252,          324;
  1:        343,        2646,         6804,          5832;
  2:       2401,       24696,        95256,        163296,         104976;
  3:      16807,      216090,      1111320,       2857680,        3674160,         1889568;
  4:     117649,     1815156,     11668860,      40007520,       77157360,        79361856,        34012224;
  5:     823543,    14823774,    114354828,     490092120,     1260236880,      1944365472,      1666598976,       612220032;
  6:    5764801,   118590192,   1067311728,    5489031744,    17643316320,     36294822144,     46664771328,     34284321792,     11019960576;
  7:   40353607,   933897762,   9605805552,   57634833312,   222305785632,    571643448768,    979960197888,   1079956136448,    694257516288,    198359290368;
  8:  282475249,  7263649260,  84050798580,  576348333120,  2593567499040,   8003008282752,  17149303463040,  25198976517120,  24299013070080,  13885150325760,   3570467226624;
  9: 1977326743, 55930099302, 719101276740, 5547352706280, 28529242489440, 102705272961984, 264099273330816, 485080297954560, 623674668798720, 534578287541760, 274925976450048, 64268410079232;
%F Formula
The row polynomials satisfy P(n,x) = (7+18*x)*P(n-1,x) for n >= 1, with P(0,x) = 49 + 252*x + 324*x^2.
%t Mathematica
P[0, x_] := 49 + 252*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[(7 + 18*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] = 49 + 252*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = (7 + 18*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.