Real-rooted atlas

A145324 → basis A007318 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

\(P_{n}(x) = \left(3 + 2\,x + 2\,n + 2\,n x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1549
2355824117
33158026801921989
4346511972155008912192041769
545045197216345164301856131904230401044225
66756753588870793848493601366204544219225632256030282525
71148647571821590192376148286138056255239424136541056405596165160960999323325

All 15 rows

Technical details
Node id
n-41376bd2608e2d0a · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-d0215f31c51c51c9

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:          5,           4;
  2:         35,          58,           24;
  3:        315,         802,          680,          192;
  4:       3465,       11972,        15500,         8912,         1920;
  5:      45045,      197216,       345164,       301856,       131904,        23040;
  6:     675675,     3588870,      7938484,      9360136,      6204544,      2192256,       322560;
  7:   11486475,    71821590,    192376148,    286138056,    255239424,    136541056,     40559616,     5160960;
  8:  218243025,  1571366760,   4947935432,   8899393728,  10000034064,   7188589696,   3228371712,   828131328,    92897280;
  9: 4583103525, 37363562460, 135333979272, 285845976928, 387988589904, 350961064896, 211567599872, 81958192128, 18513469440, 1857945600;
%F Formula
The row polynomials satisfy P(n,x) = (2*n+3+(2*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[(2*n + 3 + (2*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] = (2*n + 3 + (2*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

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.