Real-rooted atlas

A088996 → basis A013609 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1347
2215232105
32318249763842415
43465160562782421376614474865
5658353743648497769626245442561228802919735
6151420510190412285295844253497635620864158883842949120137227545
7408835353175388641055630304194727270421527426561426370560524500992825753607547514975

All 15 rows

Technical details
Node id
n-af792834b5732dff · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-c3f9a818c9640494

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:           3,            4;
  2:          21,           52,            32;
  3:         231,          824,           976,           384;
  4:        3465,        16056,         27824,         21376,          6144;
  5:       65835,       374364,        849776,        962624,        544256,        122880;
  6:     1514205,     10190412,      28529584,      42534976,      35620864,      15888384,       2949120;
  7:    40883535,    317538864,    1055630304,    1947272704,    2152742656,    1426370560,     524500992,      82575360;
  8:  1267389585,  11151977904,   42885783072,   94145623552,  129047748864,  113105252352,   61903388672,   19343867904,   2642411520;
  9: 44358635475, 435945251700, 1902473612064, 4838985014912, 7905913658112, 8604402791424, 6238407688192, 2905557368832, 788863647744, 95126814720;
%F Formula
The row polynomials satisfy P(n,x) = (4*n-1+4*n*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 - 1 + 4*n*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 - 1 + 4*n*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

Candidate equalities ?

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.