Real-rooted atlas

Veronese parent of A098172 → (θ+1)²

nonnegative coefficients

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

Recurrence

\(\left(4 + 14\,x + 16\,x^{2} - 8\,n - 27\,n x - 24\,n x^{2} + 4\,n^{2} + 12\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-x^{3} + x^{5} - 3\,n x^{3} - 9\,n x^{4} - 6\,n x^{5} + 4\,n^{2} x^{2} + 16\,n^{2} x^{3} + 21\,n^{2} x^{4} + 9\,n^{2} x^{5}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 4\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22Pn(1)
1044
2000162541
300000369864198
4000000064243300121728
50000000001004848646761962320
6000000000001448451960225012802896768
700000000000001961350384057804860216640018592
80000000000000002562023680412635140009261338852948896

All 15 rows

Technical details
Node id
n-5d6e0e5bf7e20b4a · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-cfa99f98cd44f002

Observed on the stored rows ?

  • Degrees: 1, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, 34, 37, 40, 43 (affine: slope 3, intercept -2)
  • 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 = 1, …, ascending powers of x
%e Example
Triangle begins:
   1: 0, 4;
   2: 0, 0, 0, 16, 25;
   3: 0, 0, 0,  0,  0, 36, 98, 64;
   4: 0, 0, 0,  0,  0,  0,  0, 64, 243, 300, 121;
   5: 0, 0, 0,  0,  0,  0,  0,  0,   0, 100, 484, 864, 676,  196;
   6: 0, 0, 0,  0,  0,  0,  0,  0,   0,   0,   0, 144, 845, 1960, 2250, 1280,  289;
   7: 0, 0, 0,  0,  0,  0,  0,  0,   0,   0,   0,   0,   0,  196, 1350, 3840, 5780, 4860,  2166,   400;
   8: 0, 0, 0,  0,  0,  0,  0,  0,   0,   0,   0,   0,   0,    0,    0,  256, 2023, 6804, 12635, 14000,  9261,  3388,   529;
   9: 0, 0, 0,  0,  0,  0,  0,  0,   0,   0,   0,   0,   0,    0,    0,    0,    0,  324,  2888, 11200, 24696, 33880, 29624, 16128,  5000,   676;
  10: 0, 0, 0,  0,  0,  0,  0,  0,   0,   0,   0,   0,   0,    0,    0,    0,    0,    0,     0,   400,  3969, 17424, 44436, 72576, 78750, 56784, 26244, 7056, 841;
%F Formula
The row polynomials satisfy (4*n^2-8*n+4+(12*n^2-27*n+14)*x+(9*n^2-24*n+16)*x^2)*P(n,x) = (4*n^2*x^2+(16*n^2-3*n-1)*x^3+(21*n^2-9*n)*x^4+(9*n^2-6*n+1)*x^5)*P(n-1,x) for n >= 2, with P(1,x) = 4*x.
%t Mathematica
P[1, x_] := 4*x;
P[n_, x_] := P[n, x] = Expand[Together[((4*n^2*x^2 + (16*n^2 - 3*n - 1)*x^3 + (21*n^2 - 9*n)*x^4 + (9*n^2 - 6*n + 1)*x^5)*P[n - 1, x])/(4*n^2 - 8*n + 4 + (12*n^2 - 27*n + 14)*x + (9*n^2 - 24*n + 16)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 4*x; for(n = 2, N, my(i = n); v[i] = ((4*n^2*x^2 + (16*n^2 - 3*n - 1)*x^3 + (21*n^2 - 9*n)*x^4 + (9*n^2 - 6*n + 1)*x^5)*v[i - 1])/(4*n^2 - 8*n + 4 + (12*n^2 - 27*n + 14)*x + (9*n^2 - 24*n + 16)*x^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

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.