Real-rooted atlas

basis-map coefficients of A030528, shifted → basis A013610 → (θ+1)²

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

Recurrence

\(\left(1 - 3\,x + 9\,n x + 9\,n^{2} x^{2}\right)\,P_{n}(x) = \left(-1 - 9\,x - 27\,x^{2} - 27\,x^{3} - 9\,n x - 45\,n x^{2} - 54\,n x^{3} - 9\,n^{2} x^{2} - 27\,n^{2} x^{3}\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)

Rows ?

nx0x1x2x3x4x5x6x7Pn(1)
011
1-1-12-13
212481106
3-1-36-243-432-712
4148486172820254288
5-1-60-810-4320-10125-8748-24064
617212158640303755248835721128512
7-1-84-1701-15120-70875-183708-250047-139968-661504

All 15 rows

Technical details
Node id
n-6a7ca743a949bcf0 · revision 0 · JSON
Definition source
derived:theta_plus_one_squared
Fingerprint
f6-10c18c8945c6e24a

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): no
  • 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,  -12;
  2:  1,   24,    81;
  3: -1,  -36,  -243,   -432;
  4:  1,   48,   486,   1728,    2025;
  5: -1,  -60,  -810,  -4320,  -10125,    -8748;
  6:  1,   72,  1215,   8640,   30375,    52488,    35721;
  7: -1,  -84, -1701, -15120,  -70875,  -183708,  -250047,  -139968;
  8:  1,   96,  2268,  24192,  141750,   489888,  1000188,  1119744,   531441;
  9: -1, -108, -2916, -36288, -255150, -1102248, -3000564, -5038848, -4782969, -1968300;
%F Formula
The row polynomials satisfy (1+(9*n-3)*x+9*n^2*x^2)*P(n,x) = -(1+(9*n+9)*x+(9*n^2+45*n+27)*x^2+(27*n^2+54*n+27)*x^3)*P(n-1,x) for n >= 1, with P(0,x) = 1.
%t Mathematica
P[0, x_] := 1;
P[n_, x_] := P[n, x] = Expand[Together[(-(1 + (9*n + 9)*x + (9*n^2 + 45*n + 27)*x^2 + (27*n^2 + 54*n + 27)*x^3)*P[n - 1, x])/(1 + (9*n - 3)*x + 9*n^2*x^2)]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (-(1 + (9*n + 9)*x + (9*n^2 + 45*n + 27)*x^2 + (27*n^2 + 54*n + 27)*x^3)*v[i - 1])/(1 + (9*n - 3)*x + 9*n^2*x^2)); 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

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.