Real-rooted atlas

A088996 → basis A007318 → θ+1

nonnegative coefficients

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

Recurrence

\(\left(-1 + n\right)\,P_{n}(x) = \left(-9\,n - 6\,n x - n x^{2} + 12\,n^{2} + 10\,n^{2} x + 2\,n^{2} x^{2} - 4\,n^{3} - 4\,n^{3} x - n^{3} x^{2}\right)\,P_{n-2}(x) + \left(1 - 6\,n - 2\,n x + 4\,n^{2} + 2\,n^{2} x\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 1 + 2\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1123
2310619
315687524182
41055969335681202322
594564141286711332463072036908
61039581894199263227588135920412565040701356
7135135121015234503064818432375835516780564024444032015493200
82027025203144406627641410907974410455964561251048216985164284288362880389854000
9344594253818319301400943978264967261630046610852170511982101201077629601763249547376362880011003285600
106547290757943995170323454145326902303274490209468315772956606784455084453417190172448427159850461947360039916800344144306400
1113749310575181227938220810329625477192388310076028434655348452813973855996192752871391592106138126430243195262865217371760833388336047900160011811681670400
123162341432254498226032860218996842739315721458532511294257953812835105667302389676837287944039894761909127419219390263557508553242558552010525486245121201151462406227020800441303451590400
1379058535806251206777385453506352075144890451809959160542604328618585685394541121066392221263695840613009811243444794157120811811857988227584183922243575601054274014857361792993187635218472971628808717829120017827201149811200
1421345804667687534796628409819501968483540065656560726104271779148120401303444552085166234801654143678166952104991892955124863544230579592702345716479609922967059137267480092897800952212962094272110625472321814323522432302053710873601307674368000774198357476723200
Technical details
Node id
n-a80c47bb97a26491 · revision 0 · JSON
Definition source
derived:theta_plus_one
Fingerprint
f6-e0c442c44ccf7baf

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:        1,         2;
  2:        3,        10,          6;
  3:       15,        68,         75,         24;
  4:      105,       596,        933,        568,        120;
  5:      945,      6414,      12867,      11332,       4630,        720;
  6:    10395,     81894,     199263,     227588,     135920,      41256,       5040;
  7:   135135,   1210152,    3450306,    4818432,    3758355,    1678056,     402444,     40320;
  8:  2027025,  20314440,   66276414,  109079744,  104559645,   61251048,   21698516,   4284288,   362880;
  9: 34459425, 381831930, 1400943978, 2649672616, 3004661085, 2170511982, 1012010776, 296017632, 49547376, 3628800;
%F Formula
The row polynomials satisfy (n-1)*P(n,x) = -(4*n^3-12*n^2+9*n+(4*n^3-10*n^2+6*n)*x+(n^3-2*n^2+n)*x^2)*P(n-2,x) + (4*n^2-6*n+1+(2*n^2-2*n)*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1 + 2*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 1 + 2*x;
P[n_, x_] := P[n, x] = Expand[Together[(-(4*n^3 - 12*n^2 + 9*n + (4*n^3 - 10*n^2 + 6*n)*x + (n^3 - 2*n^2 + n)*x^2)*P[n - 2, x] + (4*n^2 - 6*n + 1 + (2*n^2 - 2*n)*x)*P[n - 1, x])/(n - 1)]];
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] = 1 + 2*x; for(n = 2, N, my(i = n + 1); v[i] = (-(4*n^3 - 12*n^2 + 9*n + (4*n^3 - 10*n^2 + 6*n)*x + (n^3 - 2*n^2 + n)*x^2)*v[i - 2] + (4*n^2 - 6*n + 1 + (2*n^2 - 2*n)*x)*v[i - 1])/(n - 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. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.