Real-rooted atlas

A038321 → basis A013611 → reverse

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(28 + 18\,x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 784 + 1008\,x + 324\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9Pn(1)
078410083242116
1219524233627216583297336
2614656158054415240966531841049764477456
317210368553190407112448045722880146966401889568205962976
44818903041858719744298722816025604812801234517760317447424340122249474296896
5134929285126071817830411709934387212546358272080655160320311098475526666395904612220032435817657216
6377801998336194298170572843717088378885620768505856451668897792023228686172167466363412481371372871681101996057620047612231936
7105784559534086120392373043215738151816396823607227724595222764112448716814634072288460862717452664832172792981831682777030065152198359290368922190162669056

All 15 rows

Technical details
Node id
n-8c08995d48487910 · revision 0 · JSON
Definition source
derived:reversal
Fingerprint
f6-9111d60e4a92b32c

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:              784,              1008,                324;
  1:            21952,             42336,              27216,               5832;
  2:           614656,           1580544,            1524096,             653184,             104976;
  3:         17210368,          55319040,           71124480,           45722880,           14696640,            1889568;
  4:        481890304,        1858719744,         2987228160,         2560481280,         1234517760,          317447424,           34012224;
  5:      13492928512,       60718178304,       117099343872,       125463582720,        80655160320,        31109847552,         6666395904,          612220032;
  6:     377801998336,     1942981705728,      4371708837888,      5620768505856,      4516688977920,      2322868617216,       746636341248,       137137287168,       11019960576;
  7:   10578455953408,    61203923730432,    157381518163968,    236072277245952,    227641124487168,    146340722884608,     62717452664832,     17279298183168,     2777030065152,     198359290368;
  8:  296196766695424,  1904122071613440,   5508353135738880,   9442891089838080,  10623252476067840,   8195080481538048,   4390221686538240,   1612734497095680,   388784209121280,   55540601303040,    3570467226624;
  9: 8293509467471872, 58646959805693952, 188508085089730560, 363551306958766080, 467423108946984960, 420680798052286464, 270437655890755584, 124180556276367360, 39915178803118080, 8553252600668160, 1099703905800192, 64268410079232;
%F Formula
The row polynomials satisfy P(n,x) = (28+18*x)*P(n-1,x) for n >= 1, with P(0,x) = 784 + 1008*x + 324*x^2.
%t Mathematica
P[0, x_] := 784 + 1008*x + 324*x^2;
P[n_, x_] := P[n, x] = Expand[(28 + 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] = 784 + 1008*x + 324*x^2; for(n = 1, N, my(i = n + 1); v[i] = (28 + 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.