Real-rooted atlas

A094638 → basis A013610 → basis A013609

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
15611
24511472231
35852292298812967161
49945530041058049374431104293601
5208845141143438120045142744346550493312014973651
652211254280427014611172426580074427177638414808614433592320913392711
7151412625146061108060350193361384491398419045146384157091063047193795328141087744064850882481

All 15 rows

Technical details
Node id
n-41692a187daca5b0 · revision 0 · JSON
Definition source
derived:basis_map_A013609
Fingerprint
f6-fb5c1e44a87f37f6

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,             6;
  2:           45,           114,             72;
  3:          585,          2292,           2988,           1296;
  4:         9945,         53004,         105804,          93744,          31104;
  5:       208845,       1411434,        3812004,        5142744,        3465504,          933120;
  6:      5221125,      42804270,      146111724,      265800744,      271776384,       148086144,        33592320;
  7:    151412625,    1460611080,     6035019336,    13844913984,    19045146384,     15709106304,      7193795328,     1410877440;
  8:   4996616625,   55467971640,   269264969928,   746563089600,  1293045701904,   1432567534464,    991432348416,   391861131264,    67722117120;
  9: 184874815125, 2322132248430, 12958074355896, 42163142691312, 88157097808848, 122829466677984, 114041643752448, 68036208671232, 23666219421696, 3656994324480;
%F Formula
The row polynomials satisfy P(n,x) = (4*n+1+6*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 + 6*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 + 6*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

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.