Real-rooted atlas

A225471 → basis A013610 → basis A007318

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(3 + 3\,x + 4\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
17310
277549140
311551041297272520
42194523244876614048155440
55047356004472713505859060752431441440
61362784517726274912779123959803397952478672943243200
74224631955903980293361403431016587531772158517877519695721871470268800

All 15 rows

Technical details
Node id
n-4d7928aff3cf47a9 · revision 0 · JSON
Definition source
derived:shift_x_plus_one
Fingerprint
f6-4261b7b710db735e

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:            7,            3;
  2:           77,           54,            9;
  3:         1155,         1041,          297,           27;
  4:        21945,        23244,         8766,         1404,          81;
  5:       504735,       600447,       271350,        58590,        6075,        243;
  6:     13627845,     17726274,      9127791,      2395980,      339795,      24786,       729;
  7:    422463195,    590398029,    336140343,    101658753,    17721585,    1787751,     96957,     2187;
  8:  14786211825,  21931320600,  13536106092,   4566477384,   925231734,  115736040,   8756748,   367416,    6561;
  9: 576662261175, 899680138875, 593702099388, 218700936252, 49783469778, 7289400762, 688721292, 40599468, 1358127, 19683;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+3+3*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[(4*n + 3 + 3*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] = (4*n + 3 + 3*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

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.