Real-rooted atlas

A225470 → x ↦ x+1 → D+1

nonnegative coefficients

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

Recurrence

\(P_{n}(x) = \left(-9 + 6\,x - x^{2} + 18\,n - 6\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-3 + 2\,x + 6\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 4 + x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1415
22711139
3261135211418
4329419804053415714
55135434344837094550195064
69535326917401910792614518906911864456
72054905215908724483594375996967410340291142124592
8504199728411885000135162432236234882430729151956567011611077459120
9138779321761186475493641473830607893466209107154066560133099608910144130777463360
1042348053232037655594539213885094163628355344380358693879529587704316209963585090133651751971142388160
111418913457776013059030251376504231323151610929688826761492152665851357008057984358543536007983103801519305209133547112941440
125178799861689604914322168354561975528251075604507979124448865653111412466474626884444458008283321688240867425861317503202702724611259204418129280
132045188054989504019947055339615104831004905601634419834967386830963054388598040783226703140861224188837973301308473942495125616496144026883282882636855286151032742579123200
148689068655785100808687522028807060483736806443286207849281641888727208015007926333483720168254245432072813558994357208380129159446053501740127601129417789526529703244103154914032912220990565060377600
Technical details
Node id
n-5d8a918ab44350bc · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-7b72eafb4dd70434

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): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           1;
  1:           4,           1;
  2:          27,          11,          1;
  3:         261,         135,         21,         1;
  4:        3294,        1980,        405,        34,        1;
  5:       51354,       34344,       8370,       945,       50,       1;
  6:      953532,      691740,     191079,     26145,     1890,      69,      1;
  7:    20549052,    15908724,    4835943,    759969,    67410,    3402,     91,    1;
  8:   504199728,   411885000,  135162432,  23623488,  2430729,  151956,   5670,  116,   1;
  9: 13877932176, 11864754936, 4147383060, 789346620, 91071540, 6656013, 309960, 8910, 144, 1;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n^2-18*n+9+(6*n-6)*x+x^2)*P(n-2,x) + (6*n-3+2*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 4 + x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 4 + x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 18*n + 9 + (6*n - 6)*x + x^2)*P[n - 2, x] + (6*n - 3 + 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; v[2] = 4 + x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 18*n + 9 + (6*n - 6)*x + x^2)*v[i - 2] + (6*n - 3 + 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.