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

\(\left(9 + 28\,x + 25\,x^{2} - 12\,n - 37\,n x - 30\,n x^{2} + 4\,n^{2} + 12\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(x^{2} + 4\,x^{3} + 7\,x^{4} + 4\,x^{5} - 4\,n x^{2} - 17\,n x^{3} - 25\,n x^{4} - 12\,n x^{5} + 4\,n^{2} x^{2} + 16\,n^{2} x^{3} + 21\,n^{2} x^{4} + 9\,n^{2} x^{5}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21Pn(1)
111
20091625
30000257249146
400000049192243100584
500000000814007265761691952
600000000001217201690196011252565872
70000000000001691176337551204335194436116480
8000000000000002251792606911340126358400308748444032

All 15 rows

Technical details
Node id
n-9e444e7404c8ff14 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-7d7bdc1257ffa7e8

Observed on the stored rows ?

  • Degrees: 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42 (affine: slope 3, intercept -3)
  • 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: 1;
   2: 0, 0, 9, 16;
   3: 0, 0, 0,  0, 25, 72, 49;
   4: 0, 0, 0,  0,  0,  0, 49, 192, 243, 100;
   5: 0, 0, 0,  0,  0,  0,  0,   0,  81, 400, 726, 576,  169;
   6: 0, 0, 0,  0,  0,  0,  0,   0,   0,   0, 121, 720, 1690, 1960, 1125,  256;
   7: 0, 0, 0,  0,  0,  0,  0,   0,   0,   0,   0,   0,  169, 1176, 3375, 5120, 4335,  1944,   361;
   8: 0, 0, 0,  0,  0,  0,  0,   0,   0,   0,   0,   0,    0,    0,  225, 1792, 6069, 11340, 12635,  8400,  3087,   484;
   9: 0, 0, 0,  0,  0,  0,  0,   0,   0,   0,   0,   0,    0,    0,    0,    0,  289,  2592, 10108, 22400, 30870, 27104, 14812,  4608,   625;
  10: 0, 0, 0,  0,  0,  0,  0,   0,   0,   0,   0,   0,    0,    0,    0,    0,    0,     0,   361,  3600, 15876, 40656, 66654, 72576, 52500, 24336, 6561, 784;
%F Formula
The row polynomials satisfy (4*n^2-12*n+9+(12*n^2-37*n+28)*x+(9*n^2-30*n+25)*x^2)*P(n,x) = ((4*n^2-4*n+1)*x^2+(16*n^2-17*n+4)*x^3+(21*n^2-25*n+7)*x^4+(9*n^2-12*n+4)*x^5)*P(n-1,x) for n >= 2, with P(1,x) = 1.
%t Mathematica
P[1, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(((4*n^2 - 4*n + 1)*x^2 + (16*n^2 - 17*n + 4)*x^3 + (21*n^2 - 25*n + 7)*x^4 + (9*n^2 - 12*n + 4)*x^5)*P[n - 1, x])/(4*n^2 - 12*n + 9 + (12*n^2 - 37*n + 28)*x + (9*n^2 - 30*n + 25)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 1; for(n = 2, N, my(i = n); v[i] = (((4*n^2 - 4*n + 1)*x^2 + (16*n^2 - 17*n + 4)*x^3 + (21*n^2 - 25*n + 7)*x^4 + (9*n^2 - 12*n + 4)*x^5)*v[i - 1])/(4*n^2 - 12*n + 9 + (12*n^2 - 37*n + 28)*x + (9*n^2 - 30*n + 25)*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.