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

\(\left(1 + 4\,x + 9\,x^{2} - 4\,n - 17\,n x - 18\,n x^{2} + 4\,n^{2} + 12\,n^{2} x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(x^{2} - x^{4} + 4\,n x^{2} + 11\,n x^{3} + 7\,n x^{4} + 4\,n^{2} x^{2} + 16\,n^{2} x^{3} + 21\,n^{2} x^{4} + 9\,n^{2} x^{5}\right)\,P_{n-1}(x)\)
\(P_{1}(x) = 9\,x^{2}\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14x15x16x17x18x19x20x21x22x23Pn(1)
10099
20000253661
30000004912881258
40000000081300363144888
5000000000012157610147842252720
60000000000001699802250256014453247728
7000000000000002251536433564805415240044120832
8000000000000000028922687581140001543510164370357654016

All 15 rows

Technical details
Node id
n-23dd1c780661a15d · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-d772db5482e65522

Observed on the stored rows ?

  • Degrees: 2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44 (affine: slope 3, intercept -1)
  • 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, 0, 9;
   2: 0, 0, 0, 0, 25, 36;
   3: 0, 0, 0, 0,  0,  0, 49, 128, 81;
   4: 0, 0, 0, 0,  0,  0,  0,   0, 81, 300, 363, 144;
   5: 0, 0, 0, 0,  0,  0,  0,   0,  0,   0, 121, 576, 1014, 784,  225;
   6: 0, 0, 0, 0,  0,  0,  0,   0,  0,   0,   0,   0,  169, 980, 2250, 2560, 1445,  324;
   7: 0, 0, 0, 0,  0,  0,  0,   0,  0,   0,   0,   0,    0,   0,  225, 1536, 4335, 6480, 5415,  2400,   441;
   8: 0, 0, 0, 0,  0,  0,  0,   0,  0,   0,   0,   0,    0,   0,    0,    0,  289, 2268, 7581, 14000, 15435, 10164,  3703,   576;
   9: 0, 0, 0, 0,  0,  0,  0,   0,  0,   0,   0,   0,    0,   0,    0,    0,    0,    0,  361,  3200, 12348, 27104, 37030, 32256, 17500,  5408,   729;
  10: 0, 0, 0, 0,  0,  0,  0,   0,  0,   0,   0,   0,    0,   0,    0,    0,    0,    0,    0,     0,   441,  4356, 19044, 48384, 78750, 85176, 61236, 28224, 7569, 900;
%F Formula
The row polynomials satisfy (4*n^2-4*n+1+(12*n^2-17*n+4)*x+(9*n^2-18*n+9)*x^2)*P(n,x) = ((4*n^2+4*n+1)*x^2+(16*n^2+11*n)*x^3+(21*n^2+7*n-1)*x^4+9*n^2*x^5)*P(n-1,x) for n >= 2, with P(1,x) = 9*x^2.
%t Mathematica
P[1, x_] := 9*x^2;
P[n_, x_] := P[n, x] = Expand[Together[(((4*n^2 + 4*n + 1)*x^2 + (16*n^2 + 11*n)*x^3 + (21*n^2 + 7*n - 1)*x^4 + 9*n^2*x^5)*P[n - 1, x])/(4*n^2 - 4*n + 1 + (12*n^2 - 17*n + 4)*x + (9*n^2 - 18*n + 9)*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 1, 10}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N)); v[1] = 9*x^2; for(n = 2, N, my(i = n); v[i] = (((4*n^2 + 4*n + 1)*x^2 + (16*n^2 + 11*n)*x^3 + (21*n^2 + 7*n - 1)*x^4 + 9*n^2*x^5)*v[i - 1])/(4*n^2 - 4*n + 1 + (12*n^2 - 17*n + 4)*x + (9*n^2 - 18*n + 9)*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.