Real-rooted atlas

Veronese parent of A098172 → D

nonnegative coefficientsonly rational roots

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

Recurrence

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

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23x24x25x26x27x28x29x30x31x32x33x34x35x36x37x38x39x40x41Pn(1)
10
20235
3000410620
40000062124960
50000000836604412160
600000000010551201307015400
700000000000127821030024010218960
8000000000000014105336595630399140212240
900000000000000016136504106414001176616184245120
10000000000000000001817172017642772289820169002342711520
110000000000000000000202109902760504063005460324012602903025600
120000000000000000000002225313204125858012474129369570495017053523356320
1300000000000000000000000243001716594013860229682772024552158407260224442036122880
140000000000000000000000000263512184829421450398975491256628437582502510296288649439266240
15000000000000000000000000000284062730112843203266066102102120120108108740743803814196364057442573440
Technical details
Node id
n-2224fadf43dd48ee · revision 0 · JSON
Definition source
derived:derivative
Fingerprint
f6-8a8f26bfa265b5f7

Observed on the stored rows ?

  • Degrees: -1, 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41 (affine: slope 3, intercept -4)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 0;
   2: 0, 2, 3;
   3: 0, 0, 0, 4, 10, 6;
   4: 0, 0, 0, 0,  0, 6, 21, 24,  9;
   5: 0, 0, 0, 0,  0, 0,  0,  8, 36, 60, 44,  12;
   6: 0, 0, 0, 0,  0, 0,  0,  0,  0, 10, 55, 120, 130,  70,  15;
   7: 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,  12,  78, 210, 300, 240, 102,  18;
   8: 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,  14, 105, 336, 595, 630,  399,  140,   21;
   9: 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,   0,   0,  16, 136, 504, 1064, 1400, 1176,  616,  184,   24;
  10: 0, 0, 0, 0,  0, 0,  0,  0,  0,  0,  0,   0,   0,   0,   0,   0,   0,  18,  171,  720, 1764, 2772, 2898, 2016, 900, 234, 27;
%F Formula
The row polynomials satisfy (n-2)*P(n,x) = ((n-1)*x^2+(n-1)*x^3)*P(n-1,x) for n >= 3, with P(1,x) = 0, P(2,x) = 2*x + 3*x^2.
%t Mathematica
P[1, x_] := 0;
P[2, x_] := 2*x + 3*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((n - 1)*x^2 + (n - 1)*x^3)*P[n - 1, x])/(n - 2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 0; v[2] = 2*x + 3*x^2; for(n = 3, N, my(i = n); v[i] = (((n - 1)*x^2 + (n - 1)*x^3)*v[i - 1])/(n - 2)); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i, v[i])))};
T(10)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

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.