Real-rooted atlas

A109692 → basis A013609 → basis A007318

nonnegative coefficientsonly rational roots

Proved for all n all roots real, in (−∞, −1]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(-2 - 2\,x + 6\,n + 4\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1426
240441296
364011046321202496
414080332482936011488168089856
539424011843841420544850144253824302404133376
613404160489423367435494460156864273331846612288665280231469056
753616640023062016004246698496433950310426574058249751543041985306881729728015276957696

All 15 rows

Technical details
Node id
n-8589eb5ac049f4cd · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-ca0eccf426952f5b

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • 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:             1;
  1:             4,             2;
  2:            40,            44,             12;
  3:           640,          1104,            632,            120;
  4:         14080,         33248,          29360,          11488,           1680;
  5:        394240,       1184384,        1420544,         850144,         253824,          30240;
  6:      13404160,      48942336,       74354944,       60156864,       27333184,        6612288,        665280;
  7:     536166400,    2306201600,     4246698496,     4339503104,     2657405824,      975154304,     198530688,      17297280;
  8:   24663654400,  122170265600,   264534178816,   327018097664,   252425761024,   124579272704,   38387040768,    6751595520,    518918400;
  9: 1282510028800, 7191418060800, 17909566328832, 25999103158272, 24244754893824, 15060598055424, 6231821391872, 1656242353152, 256538004480, 17643225600;
%F Formula
The row polynomials satisfy P(n,x) = (6*n-2+(4*n-2)*x)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[(6*n - 2 + (4*n - 2)*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] = 1; for(n = 1, N, my(i = n + 1); v[i] = (6*n - 2 + (4*n - 2)*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.